All [ Hardy's ] books gave him some degree of pleasure,
but this one, his last, was his favourite. [ ... ] He believed in its value (as well he might). Preface, byJohn E. Littlewood, to the
last book of G.H. Hardy
(1877-1947): "Divergent Series" (1949)
Being eternal, Logic can be patient.Pierre Duhem (1913)
In 1713, Leibniz said that all divergent series
can be evaluated. Can they?
A series is just an ordered sequence of infinitely many terms which we seek to add together
(order matters).
A partial sum of such a series is obtained by adding its first terms.
If partial sums have a limit, the series is said to be convergent and that limit is called the sum of
the series. Non-convergent series are called divergent.
The notion of sum can be extented to some divergent series (either stable
or unstable in the sense discussed below).
(2012-07-24)
Sum of a Divergent Geometric Series
How can a definite sum be assigned to a divergent series?
Pour moi, j'avoue que tous les raisonnements
fondés sur les séries qui ne sont pas convergentes [...]
me paraîtront très suspects, même quand les résultats s'accorderaient
avec des vérités connues d'ailleurs.
D'Alembert, 1768
[Opusc. math., 5, p. 183 ]
Analytic continuations
can make sense of some divergent series in a consistent way.
Consider the following
classic summation formula
(attributed to Eudoxus) for the
geometric series, which converges when the common ratio
z is small enough in
magnitude (it diverges when |z| > 1 ) :
1 + z +
z2 +
z3 +
z4 + ... +
zn + ...
= 1 / (1-z)
The right-hand-side always makes sense, unless z = 1.
It's tempting to equate it formally
to the left-hand-side, even when that series diverges!
This viewpoint belongs to a consistent body of knowledge which is still
not mature, in spite of its exploration by generations of great mathematicians.
The following monstrosities do make sense as "sums" of divergent series :
In rings, whenever both sides of such equations are defined,
they are necessarily equal. In p-adic
arithmetic, for example, the above geometric series
converges (only) when z is an integer
which is divisible by the modulus p
(use p=2 and p=3, respectively, for the above two examples).
Convergence in some ad hoc realm is a
comforting luxury which is rarely available.
There's no such thing in the case of what's arguably
the simplest divergent series
(Grandi's series, 1703)
whose long history started with an incidental remark of
Leibniz in 1674
(De Triangulo Harmonico) :
1 - 1 +
1 - 1 +
... + (-1)n + ...
= ½
The best way to index the successive terms in Grandi's series is to start with n = 1.
That makes it a multiplicative sequence.
The correct way to extend any multiplicative sequence
down to the index n = 0 (when needed) is to assign it a value of zero there.
Thus, I consider dubious the popular use
of A033999
as a representation of Grandi's series. I'm adamant about using instead the following
sequence (whose closed form on the left-hand-side depends on the
fact that the zeroth power of zero is unity).
Because Grandi's series is stable
(as a geometric series whose ratio isn't unity)
the leading zero is irrelevant to the summation.
This indexation which makes Grandi's sequence multiplicative also makes its
Dirichlet inverse multiplicative
(and characterized by values at powers of primes):
(2018-05-23)
Definitions and notations (inherited from tradition).
The equal sign may be used to say that a scalar is the sum of a series.
A (formal) series is just a
vectorial object
consisting of an infinite sequence of scalar coefficients
(called terms).
The formal series of general term an can be denoted:
Snan
A summation method (also summation mapping or
summation, for short)
is a recipe which assigns a scalar value (called sum) to some
series in such a way that the newly defined sum
of a series with only finitely many nonzero terms is the ordinary (finite) sum of those
nonzero terms.
The sum of a series using the summation method M is denoted with the name
of M above the sigma sign (or as a superscript). If the summation method is clear from
the context, square brackets may also be used (such bracket thus turn a series into a scalar):
M
[ Snan ]
=
Sn
an
For typographical simplicity and for compatibility with traditional notations, brackets
may be dropped in equations when it's clear that summation is intended (because the vectorial object itself wouldn't
make sense in the context). That's so when a series is equated to an explicit scalar. Example:
[ Snan ]
=
Snan
In particular, the above requirements for any summation method imply:
0 = Sn 0
Normally, the index n runs from 0 to infinity,
through the set of natural integers = {0,1,2,3,...}.
It's sometimes convenient to the use the set of counting numbers instead
* = {1,2,3,4,...).
When we do so, we signal it with a star after the index
(this is not a standard notation). In other words:
Sn*an
=
Snan+1
That's just an equivalence of notation, without a deep mathematical content:
It just says that a sequence of scalars remains the same no matter how we index it.
The above just expresses the identity between two ways to denote the same formal vectorial object
irrespective of any property attached to it (including summability).
Formal series are just vectors with a (countable) infinity
of coordinates. Such vectors are equal if and only if all of their coordinates are
pairwise identical, as is the case with only finitely many
dimensions.
Another notation for that same vectorial identity is:
S
an =
S
an+1
n=1
n=0
On the other hand, if we put anything above the
sigma sign, then some kind of summation is implied.
This could be via one of the highbrow methods discussed on the present page
(or elsewhere) or by straight
convergence (when applicable)
in which case the infinity symbol
(¥)
can be used.
That way, we retrieve a familiar notation exemplified by:
¥
1 / 2n = 2
S
n=0
Thus, the infinity symbol takes the place we may use to identify
other summation methods which gives new meaning to the
nice title of a relevant short video by
Henry Reich:
Adding Past Infinity.
Reich was just lucky that the divergent example he picked
happened to be a stable series.
If we do away with the above sigma notation (as we often will)
the stability property Reich took for granted says that:
For some series this isn't so.
Such unstable series include some physically relevant ones
which motivated Reich's
next episode, like:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...
That series isn't stable (as we shall see)
but we can still uncover a sum for it (namely, -1/12).
That sum is modified by adding finitely many zeroes "in front of it",
but in a consistent way which relies only on linearity.
To navigate this foggy land, we can only rely on precise instruments...
(2012-07-29)
Desirable Properties of General Summation Methods
Two consistent methods give the same result when both are defined.
The mapping which assigns a sum to a series ought to possess a few desirable properties,
but we must be prepared to abandon some of these (except linearity)
to allow the most powerful summation methods
(e.g., those which can handle interesting unstable divergent series).
1 - Linearity :
Let's split linearity
into two separate requirements :
Scalability :
Sn l an =
l
Snan
Additivity :
Sn
( an + bn ) =
Snan +
Snbn
Linear summation methods are thus linear forms defined over the
vector space of formal series
(in any vector space, a form is
generally defined as a function which takes a vector to a scalar).
2 - Stability (a.k.a. Translativity or Shift-Invariance ) :
Stability is the following nontrivial rule, inspired by finite sums:
Snan =
a0 +
Snan+1
That's also known as shift-invariance (explicitely required for summation by
Banach limit,
when the sequence of partial sums is almost convergent ).
An uresolved question is whether a shifted stable series is necessarily stable again.
If that's so, the above rule says that you can freely shift such a series to
the right or to the left without compromising summability
and that the sum is only affected as would be naively expected
(there's no reason to distinguish between right or left stability for
linear summation methods).
The stability property is treacherous.
It turns out that it's really a property of the series
itself and not of the summation method being used.
A great deal of confusion comes from the fact that
many summation methods are valid only
for stable series
(summation by convergence being the most obvious
example). In such a case (and only
in such a case) we say that the summation method itself
is translative.
As we shall see later on,
1 + 1 + 1 + 1 + 1 + ... = -½
If that series was stable, of sum x, the equation x = 1+x
would hold. It ain't so... There's an elementary proof that
the series can't possibly be stable under any linear summation method.
(HINT: If x is the sum of the original series and y is the sum of the
shifted series, then x-y = 1.)
3 - Regularity :
A summation method is said to be regular if it assigns
to any convergent series its ordinarysum (i.e., the limit of its partial sums).
By definition, all summation methods are finitely regular
(i.e., they all give the same result for any series with finitely many nonzero terms)
but some of those are not regular
(HINT: consider the
Riemann series theorem).
4 - Multiplicativity :
The formal series form not only a vector space but
also a ring. We'd like the summation of
series to respect that structure and may want the sum
of a Cauchy product to
equal the product of the sums of its two factors, namely:
( Snan )
( Snbn )
=
Sn
( S k ≤ nan-kbk )
The ordinary summation of convergent series is certainly
multiplicative in that sense. So is the Cesàro-summation of
Cesàro-summable series:
One example involving divergent
(but Cesàro-summable) series is the following pair of series,
(the second one is the Cauchy-square of the first):
If the limit of
fn (x) is 1 (unity) for every index n,
as x tends to x0 in some limiting process,
then we have, for the same limiting process:
Snan
= lim x [
Snanfn (x) ]
This may be summarized: The sum of the limits is the limit of the sums
(the space of formal series is endowed with
the Tychonoff topology).
As every student of Analysis ought to know,
that's not so in the restricted realm of convergent series,
as the series formed by the limits (of the corresponding terms of in a sequence
of convergent series) is not necessarily a convergent series.
Historically, the desired continuity of the summation in the realm of divergent series
was rarely evoked explicitly but it has almost always been assumed by everyone...
The traditional parlance, since 1907,
is to call the coefficients fn (x)
convergence factors whenever they allow the above right-hand side to make
sense in a relevant neighborhood of the limit point for x.
It was Charles N. Moore (1882-1967)
who coined that locution in 1907. It appears
in the title of his
Harvard doctoral dissertation
(1908).
The French equivalent (facteur de convergence)
was used systematically in a new chapter (VI) of the second edition
(1928)
of Borel's " Leçons sur les séries divergentes " (1901)
without any indication of its origin (later claimed,
in print, by Moore himself).
Charles N. Moore
For regular summation methods,
whenever the series involved on the right-hand-side are convergent and their sums have a finite limit,
that limit is the same for all factors of convergence
(this is left as an exercise for the reader) and can thus be used as a definition
of the sum of the series on the left-hand-side if it happens to be divergent.
What the semi-continuity of a summation method ultimately states is that the above equation also holds
when sums of divergent series are involved on the right-hand-side...
Moore's Theorem (2016-05-05)
The key remark, which I like to call Moore's theorem,
is that two different sequences of convergence factors which "work"
will give the same result. That's to say that the following equation holds whenever
both sides make sense with two sequences f and g
whose terms all have a limit of 1.
lim x [
Snanfn (x) ]
= lim x [
Snangn (x) ]
In 1900, Alfred Pringsheim defined a doubly-indexed sequence as having a limit
L if its terms approach L when both indices tend to infinity.
(2018-05-26)
Stability of the geometric series.
The geometric series whose common ratio is unity cannot be stable.
The geometric series of common ratio 1 is not
stable, but all other geometric series can be assigned a
precise stable sum. Let's prove that:
We assume a linear summation method powerful enough to assign a finite sum
S to the geometric series at point z. Which is to say:
1 + z +
z2 +
z3 +
z4 + ... +
zn + ...
= S
Stability would then be expressed by the following relation,
which would seem obvious to the uninitiated (it's what's often implicitly used as a
preliminary step in the derivation of the sum of convergent
geometric series).
Actually, that's what we want to prove or disprove :
0 + 1 + z +
z2 +
z3 + ... +
zn + ...
= S (?)
By definition, the sum of a series
with finitely many nonzero terms is easy:
1 + 0 + 0 + 0 + 0 + 0 +
0 + 0 + ...
= 1
As linearity is assumed, we
may subtract that from our original equation:
0 + z +
z2 +
z3 +
z4 + ... +
zn + ...
= S-1
Also by linearity, we're allowed to scale that by a factor of 1/z
(if z is nonzero) and we obtain the sum of the series we were after:
0 + 1 + z +
z2 +
z3 + ... +
zn + ...
= (S-1)/z
So, excluding the trivial case z=0,
the stability of the geometric series of common ratio z
is equivalent to the equation:
S = (S-1)/ z
This equation has no solution when z=1,
which goes to show that the geometric series of common ratio unity is not
stable. Otherwise, it implies:
S = 1 / (1-z)
That's the only stable sum a linear summation method
can assign to the geometric series of common ratio z.
An ad hoc requirement for
linear summation methods could be to impose the stability of geometric series of nonunity ratio.
We've refrained from adding that to our basic list.
Continuity seems far more fundamental, albeit less elementary,
as presented below.
(2020-05-13)
Stability of any series with vanishing terms.
Any series whose terms tend to zero is stable.
This applies even when such a series doesn't converge (one example being the
harmonic series).
This is so because the difference between the series
and the series obtained by shifting it to the right
(by one position or several, with zero padding at the beginning) is a convergent series of zero sum...
Indeed, that difference is just a telescoping
series whose partial is equal to a single term of the original series
(or the sum of several such terms if we shift by several positions).
Since we assume that those terms tend to zero, the limit of the partial sums
is zero; the difference is thus a convergent series of zero sum.
(2018-06-23) The Cauchy product of two series.
It's a stable series when at least one of the factors is stable.
A couple of classical theorems pertain to the convergence and/or summability of
the Cauchy-product of two series which have prescribed properties.
(In all of these cases, the sum of the Cauchy-product is the product of the sums.)
(Cesàro, 1890)
If two series are Cesàro-summable, so is their Cauchy-product
(and the sum of that product is the product of the two sums).
The Cauchy product of two unstable summable series can be
a summable series whose sum isn't the product of the two sums
(cf. example above).
This doesn't seem to happen when at least one of the two series is stable.
It can be useful to think of the Cauchy product as an additive convolution
to be contrasted with the mulplicative convolution
corresponding to the Dirichlet product
described in the next section. They can be defined using very similar
notations. Compare the following definition of the Cauchy product
to the parallel definition given below for the Dirichlet product.
å
f × g =
å
å
f (d) g (n-d) =
å
å
f (i) g (j) =
å
f
å
g
n ≥ 0
d ≤ n
i ≥ 0
j ≥ 0
In both cases, we may consider what happens when
all series involved are absolutely convergent.
In that case, all terms can be freely permuted and the above equality is established
by remarking that every term f (i) g (j)
appears once and only once on either side.
In other words, the sum of the product is the product of the sums.
That nice statement is not necessarily true in general
(we've already seen a counter-example).
Summing by Convolutional Inversion
(2019-07-24 15:15 PDT)
Over any unital ring, the formal series
form a unital ring (whose unity is
the series whose only nonzero term is a leading term equal to unity).
Any series whose leading term is unity has an inverse with respect to the
Cauchy product, which is often called its convolutional inverse.
When that inverse is summable and has an invertible sum,
it's tempting to believe that the inverse of that is the sum of the original series.
In some cases, we can prove that to be true...
"Sur la multiplication des séries" by Ernesto Cesàro.
Darboux's Journal, 14, pp. 114-120 (1890).
(2019-05-15 02:30 PDT)
Dirichlet Product
of Two Series
The sum of the product is the product of the sums. Or is it?
When all series involved are absolutely convergent,
their terms can be freely permuted and the above statement is established
by remarking that every term f (i) g (j)
appears once and only once in either of the following summations:
å
f *g =
å
å
f (d) g (n/d) =
å
å
f (i) g (j) =
å
f
å
g
n ≥ 1
d | n
i ≥ 1
j ≥ 1
Dirichlet Summation Method :
Any series whose first term is nonzero has a Dirichlet-inverse.
If that Dirichlet-inverse is summable, then the series is said to be Dirichlet-summable and its sum is the reciprocal
of the sum of its Dirichlet-inverse.
That implies a sum of -2 for the Moebius series
å m(n) whose inverse is:
1 + 1 + 1 + 1 + 1 + ... = -1/2
A less chancy example would be the Dirichlet-inverse of Grandi's series:
(2019-07-05 17:45 PDT)
Infinite Products
Exponentials of infinite series seem less ambiguously defined.
The situation is apparently similar to the way ambiguities are lifted
when going from the formula of Roger Cotes
to Euler's formula.
However, the full formalism of Riemann sheets
cannot be bypassed in general.
(2018-07-03) Series whose terms are themselves sums of series.
Countable additivity yields two distinct ways to obtain such sums.
One special case can be derived from the following termwise equality.
Si
Sjaibj
=
Si
[ Sjaibj ]
=
Siai [ Sjbj ]
=
[ Sjbj ]
Siai
The general relation is guaranteed if we have
absolute convergence:
[ Si
Sjaij ]
=
[ Sj
Siaij ]
Otherwise, that only gives a questionable hint.
One such hint would be that the series whose
n-th term is s0(n),
the number of divisors of n,
has a sum of ¼
(HINT: All
decimated series S0(k.k-1) have the same sum).
Likewise for any other divisor function
ss ,
where ss (n)
is defined as the sum of the s-th powers of all the positive divisors of n:
The following example shows that this cannot be applied blindly.
It's useful to investigate conditions (besides absolute convergence)
which allow this.
-1
+
1/2
+
1/22
+
1/23
+
1/24
+
1/25
+
...
=
0
0
+
-1
+
1/2
+
1/22
+
1/23
+
1/24
+
...
=
0
0
+
0
+
-1
+
1/2
+
1/22
+
1/23
+
...
=
0
0
+
0
+
0
+
-1
+
1/2
+
1/22
+
...
=
0
0
+
0
+
0
+
0
+
-1
+
1/2
+
...
=
0
0
+
0
+
0
+
0
+
0
+
-1
+
...
=
0
...
-1
-
1/2
-
1/22
-
1/23
-
1/24
-
1/25
-
...
=
-2
(2018-07-16)
Original Bernoulli Numbers
(c. 1689)
Also in the posthumous legacy of
Seki (1712).
Many aspiring algebraists, like my younger self, have rediscovered those
in the elementary context of Faulhaber's formula.
Bernouilli numbers are often found in Number Theory and
play a key rôle for infinite series via the
Euler-Maclaurin formula
and Ramanujan summation.
They're best defined through their (exponential) generating function :
z / (1 - e-z ) =
Sn Bn zn / n!
(convergent for | z | < 2p )
Everybody agrees on the values of Bernoulli numbers for even values of the index n
(some authors used to report only even values by halving the index:
A000367/A002445).
However, the sign of the only nonzero odd value ( B1 ) remains controversial.
The alternate value for B1
appearing in either
A027641/A027642
or
A027641/A141056)
was endorsed de facto by the NIST (1964)
and Wolfram Research (but not by Bernoulli, Seki,
Terry Tao... or my adamant self).
The confusion arises from a competing equally-simple generating function, obtained by changing the sign of z.
The difference between the two generating functions is given by a remarkable easy-to-prove identity,
which shows that both only have one nonzero odd coefficient and that their even coefficients coincide:
z / (1 - e-z )
-
z / (ez - 1) = z
Whenever possible, it's probably best not to take either convention for granted and
give explicitly the first terms of an expansion followed by a general expression
involving only Bernoulli numbers of even rank (see example below).
(2012-08-03)
Summation as a covector (i.e., a continuous linear form).
(In infinitely many dimensions, linear forms could be
discontinuous.)
The formal series (irrespective of their convergences)
form a sequence space.
Using Dirac's notation,
a formal series can be described as a ket :
Snan
=
Snan | n >
=
| y >
A continuous and linear
summation method is a bra < s |
Such a bra is a member of the continuous dual
of the aforementioned sequence space
(the algebraic dual consisting of all
linear forms is much larger, by the Axiom of Choice).
Formally, a regular summation can only be
equal to the following bra :
< s | =
Sn < n |
However, that expression is of no practical value,
unlike some of the following methods which describe
< s | better.
Let's define the (non-invertible) shiftoperatorÛ via:
Û | n > = | n+1 >
If | y > is stable,
we have:
< s | y > =
< s | Û | y >
Duality: The smaller the subspace, the bigger its dual.
In general, the [continuous] dual
of a [topological] vector space
consists of all [continuous] linear functions defined on it
(two such functions being equal if and only if they have the same value everywhere).
The dual of any linear space E so defined is a well-defined set (a subset of
the Cartesian squareE2 )
which is itself a linear space.
In this work we consider only continuous duals and denote
E* the [continuous] dual of E.
Furthermore, our attention is restricted to the case when E
is some subspace of the sequences of scalars.
(2012-08-10)
Summation by Convergence
The only method
Cauchy (1789-1857) would ever recognize.
The sequence of the partial sums of a series is the
sequence whose term of index n (usually starting at n = 0)
is obtained by adding the finitely many terms whose index in the
series does not exceed n.
If that sequence of partial sums converges to a limit S,
the series itself is said to be convergent (of sum S).
For convergent series (at least)
we make no typographical distinction between a series
and its sum. Thus, we express the above as:
Snan
=
[
Snan
]
=
(
m
an
)
=
¥
an
lim
S
S
m ® ¥
n = 0
n = 0
The last equation merely expresses the conventional notation for the quoted limit
of partial sums. Nothing else.
When that limit doesn't exist, Cauchy
argued that the leftmost expressions don't make sense.
Two generations before him, the great Euler
had taken the opposite view, rather freely, with great success
(Ramanujan would do the same much later).
Cauchy simply had deep concerns that the lack of explicit rules
for manipulating divergent series made any argument based on them doubtful at best.
So he decided to rule them out!
The pendulum has swung back. Nowadays, divergent series can be handled
with complete analytical rigor.
Both Cauchy and Euler would be pleased...
The following sections will trace the historical path away
from Cauchy's strict point of view, then break free completely...
Summation by convergence is just the simplest
regular summation method,
among mutually consistent ones which apply to divergent series as well.
The other such methods, including those described below, can be classified into
two broad groups:
Summations by means.
Summations by convergence factors.
Sometimes, those two types of approaches are known to be equivalent.
(2012-09-09) The Functional Analysis Approach
How to extend a continuous summation method to a larger domain.
By definition, an absolutely regular summation method is
a continuous functional defined for every absolutely convergent series
which coincides with the ordinary summation by convergence.
An absolutely regular summation method need not be
regular.
For example, a permutation of the indices always leaves unchanged
the sum of any absolutely convergent series. However,
the Riemann series theorem says that it
may change arbitrarily the sum of
other convergent series and, thus, induce a non-regular summation method.
As the limiting case of this convergent summation, as z tends to 1 on [0,1[
1 - 2 z + 3 z2
- 4 z3 + ...
+ (-1)n+1 (n+1) z n + ...
= 1 / (1+z)2
Note, however, that Euler's method isn't powerful enough to handle the nonalternating
case (corresponding to z = -1) which is one of our favorite
examples of an unstable series.
This much can be construed as a consequence of the following theorem:
Proof: This is a straight consequence of the stability
of convergent series (as implied by the notation we used for Euler's above definition,
only convergent series appear in the right-hand-side).
Indeed, the limit of the sum of two sequences is the sum of their
respective limits, when both exist.
Therefore, the following is a trivial equality between identical series :
Snan
=
Sn
ò0¥ an
tn/n! e-t dt
In 1899, Emile Borel (1871-1956)
thus proposed to define
the left-hand-side (which could be a divergent series)
by equating it to the right-hand-side of the following formula,
at least when the new series that is so formed
is a power series of t with an infinite
radius of convergence, which makes
the resulting (improper) integral converge:
Snan
=
ò0¥
( Snan
tn/n! ) e-t dt
The bracketed series on the right-hand-side clearly stands a better chance
of converging than the series on the left-hand-side.
For example, in the case of the geometric series
(an = zn ) the above integrand
becomes exp [(z-1)t] which makes the integral
on the right-hand-side converge when the real part of
z is less than one. We thus have convergence of the Borel formula
for half the plane, whereas the left-hand-side merely converges on a
disk of unit radius.
Borel summation, however, is best understood as a
way to obtain the sum of a series from the sum of another,
even if the latter is not convergent...
For example, armed with the formula for the sum of a
geometric series
(convergent of not) we can use the Borel summation to
make mincemeat of the following series which Euler
(E247, 1746)
called hypergeometric series of Wallis
(the name is obsolete; I recommend the unambiguous name Euler-Gompertz series).
He evaluated it in half a dozen
distinct consistent ways:
Some authors call this "Euler's series".
To Euler, hypergeometric numbers were just what we now call
factorials
(ironically, the latter term had been coined by
Wallis himself, in 1655).
The locution "hypergeometric series"
is now reserved for a different creation of Gauss (1812).
(2012-08-21)
Nørlund summation: Linear, stable & regular (1919)
All Nørlund means are consistent with one another.
Following Niels E. Nørlund (1919) let's consider an infinite sequence of
complex ponderation coefficients, the first of which being nonzero:
c0 ¹ 0 , c1 , c2 ,
c3 , c4 , ... , cn , ...
Let's call C the sequence of the partial sums of the corresponding series:
Cn =
c0 + c1 + c2 +
c3 + c4 + ... + cn
We impose the so-called regularity condition :
The positive series of term | cn/ Cn | is convergent.
In particular, coefficients in geometric progression are thus ruled out,
if the common ratio is greater than 1. So are coefficient sequences that grow faster than
such a geometric sequence.
For any series an with partial sums
A n = a0 + ... + an
we define:
A'n =
( c0 A n + c1 A n-1 + c2 A n-2
+ ... + cn A 0 ) / C n
This expression is called a Nørlund mean.
If A'n tends to a limit S as n tends to infinity,
then S is called the Nørlund-sum of the series an
or, equivalently, the Nørlund-limit of the sequence A n .
Remarkably, the value of S doesn't depend on the choice of the sequence of coefficients
(with the above regularity restrictions).
The question of the consistency of Nørlund and
Hausdorff methods was raised by E. Ullirich and it was answered
(in the affirmative) by
W. Jurkat and A. Peyerimhoff, in 1954.
As he was reflecting on the summation of the geometric series
in 1908, Mittag-Leffler proposed the most
widely applicable definition he could think of, at the time
(using the Gamma function) :
Snan
=
(
Sn
an
G(1+en)
)
lim
e ® 0+
For the geometric case (an = zn )
the right-hand-side converges except when z is a real greater than or equal to 1.
More generally, convergence normally occurs on a Mittag-Leffler star
consisting of all points of the complex plane, except the shadows of singular points
(i.e., wherever a singular point exists on the straight ray from zero to that point).
(2016-05-08)
Generalized Summation Methods (c. 1919)
Generalizations and early attempts at defining the sum of divergent series.
In March 1918 at the University of Washington,
Lloyd Leroy Smail
(1888-1955)
defined the sum of a series by the following equation,
whenever the right-hand-side makes sense for a sequence of functions
fn (m,x) which all tend to 1 as m
tends to infinity and x tends to a given limit point x0 :
¥
[
(
m
) ]
å
an =
lim
lim
å
an fn (m, x)
n = 0
x
m ® ¥
n = 0
He pointed out that many previously devised summation methods merely
reduce to different choices for f in the work of different authors,
including:
(2016-05-03)
The Stability Theorems
When is a series summable by convergence factors stable ?
A summation method relying on convergence factors can be formulated as
defining the sum...
(2012-08-09)
Weierstrass Summation (1842)
Analytic continuation viewed as
a summation method...
This applies not only to formal power series
outside of their disk of convergence but also when each term
of the series is an analytic function of z
indexed by n (the
zeta series being the prime example of that).
If f is subexponential and the real part of z is large enough,
then the series converges. Otherwise we can extend it by analytic continuation
(as is normally done to define the zeta function itself)
to obtain the desired sum as the value of F(0)
whenever it makes sense that way.
This is well-defined because of the
uniqueness of Analytic Continuation.
The linearity can be established by noticing that the map
from a function to its Dirichlet series is a linear one
and analytic continuation preserves linearity at every point where it makes
sense, including z=0 supposedly.
We've already noted the unstability
of the first example.
The unstability of the second one results from the aforementioned
linearity of this summation method.
Indeed, by adding or subtracting both sides of the above pair of equations, we obtain:
(2016-04-27)
Wonders of Unstable Summations
Applying linear summation methods to unstable series.
In the previous section, we saw examples of series
whose lack of stability could be demonstrated using the linearity of summation.
I'm not prepared to abandon the linearity requirement for summation methods,
even if that means giving up the "obvious" stability property.
In this section, I'll show some of the consequences in more details.
For shock value.
Let's revisit the previous argument with the following notations.
In the first line, we explicitly invoke the assumption
that a finite sum is equivalent to a series ending with
infinitely many zeroes.
We could subtract a finite sum from either one of those to obtain the sum
of a 2-decimated series, starting with an odd or an even number of zeroes.
(This would result in a seamless formula, which we're about to generalize.)
This series is convergent for |z| < 1.
It's divergent but stable everywhere else in the
complex plane, except at the pole
of the right-hand-side (z=1) where the series is unstable
of sum -½ (obtained by zeta regularization).
We'll use that series when z is a primitive k-th
root of unity to obtain, by linearity alone, the sum of the decimated
series where only every k-th term subsists (we demonstrate the technique
elsewhere for convergent series).
Our preliminary example was for k=2 (the only primitive square root of unity is z = -1).
Let's do it again for k=3, with
z = w = exp (2pi/3)
w3 = 1
1 + w + w2 = 0
The first 3 lines below correspond to
z = 1, w and w2.
The other ones are just linear combinations thereof, using the coefficients highlighted at right.
More generally, the result of rank m consists of k
times the sum S0(k,m) of the series formed by retaining the term of rank m
and every k-th term thereafter
(0 ≤ m < k).
It's obtained with w = exp (2pi/k)
by assigning a coefficient w-m q
to the equation of rank q. That yields:
k S0(k,m) = -1/2 +
k-1
w - m q/ (1-w q )
S
q=1
When all is said and done S0(k,k-1) is always equal to -1/2.
S0(k,m) = 1/2 - (m+1) / k for 0 ≤ m < k
The validity of this formula extends to larger values of m, since subtracting
one unit from the leading term of such a k-decimated series clearly lowers the sum by one.
But that can also be construed as increasing m by k units.
To summarize:
The method and the notations just exemplified for the geometric series apply unchanged to
the decimation (0 ≤ m < k) of any summable power series,
convergent or not
(HINT: multiply each of the above columns into its own coefficient).
Calling f (z) the undecimated sum, we obtain:
Decimation Formula [ f is analytic on the unit circle except at 1.
f (0) ¹ 0 ]
Sf(k,m) =
1
k
k-1
exp (- 2pi mq/k ) f ( exp (2pi q/k) )
S
q=0
Let's use that formula to decimate the series 1 + 2 + 3 + 4 + 5 + ...
knowing that Sn* n zn-1 = f (z)
is -1/12 if z=1 and 1/(1-z)2 otherwise.
S1
m=0
m=1
m=2
m=3
m=4
m=5
m=6
m=7
m=8
m=9
k=1
-1/12
-13/12
-37/12
-73/12
-1/12 - m (m+1) / 2
k=2
1/12
-1/6
-11/12
-13/6
-47/12
-37/6
1/12 - m2 / 4
k=3
1/12
1/12
-1/4
-11/12
-23/12
-13/4
1/12 - m (m-1) / 6
k=4
1/24
1/6
1/24
-1/3
-23/24
-11/6
1/24 - m (m-2) / 8
k=5
-1/60
11/60
11/60
-1/60
-5/12
-61/60
-1/60 - m (m-3) / 10
k=6
-1/12
1/6
1/4
1/6
-1/12
-1/2
k=7
-13/84
11/84
23/84
23/84
11/84
-13/84
-7/12
k=8
-11/48
1/12
13/48
1/3
13/48
1/12
-11/48
-2/3
k=9
-11/36
1/36
1/4
13/36
13/36
1/4
1/36
-11/36
-3/4
10
-23/60
-1/30
13/60
11/30
5/12
11/30
13/60
-1/30
-23/60
-5/6
S1(k,m) = -k / 12 - (m+1) (m+1-k) / 2k
The white entries (for m<k) come directly from the
decimation formula. Grey entries above the diagonal satisfy the recurrence (for p=1):
Sp(k,m+k) = Sp(k,m) - (m+k)p
This establishes, by induction, the unified formula on the bottom line.
Of particular interest is the diagonal (m = k-1) where
the sum of the decimated series is just k times the sum of the undecimated series (-1/12).
Since the remaining terms are just the successive multiples of k in this case,
we may divide by k and retrieve a series which differs from the original one
only by being stretched with k-1 zero terms before each original term.
In our previous discussion of the constant series, we had already remarked
that this type of padding always left the value (-1/2) of the sum unchanged.
To the uninitiated, that property may look more obvious than it actually is:
Stretching a series this way often doesn't change
the sum, but it may...
To decimate the series of squares 1 + 4 + 9 + 16 + 25 + ...
we use Sn* n2 zn-1 = f (z)
with f (z) = (1+z)/(1-z)3 if z¹1
and f (1) = z(-2) = 0.
S2
m = 0
m = 1
m = 2
m = 3
m = 4
m=5
m=6
m=7
m=8
m=9
k=1
0
k=2
0
0
k=3
-1/9
1/9
0
k=4
-1/4
0
1/4
0
k=5
-2/5
-1/5
1/5
2/5
0
k=6
-5/9
-4/9
0
4/9
5/9
0
k=7
-5/7
-5/7
-2/7
2/7
5/7
5/7
0
k=8
-7/8
-1
-5/8
0
5/8
1
7/8
0
k=9
-28/27
-35/27
-1
-10/27
10/27
1
35/27
28/27
0
10
-6/5
-8/5
-7/5
-4/5
0
4/5
7/5
8/5
6/5
0
11
-15/11
-21/11
-20/11
-14/11
-5/11
5/11
14/11
20/11
21/11
15/11
12
-55/36
-20/9
-9/4
-16/9
-35/36
0
35/36
16/9
9/4
20/9
13
-22/13
-33/13
-35/13
-30/13
-20/13
-7/13
7/13
20/13
30/13
35/13
14
-13/7
-20/7
-22/7
-20/7
-15/7
-8/7
0
8/7
15/7
20/7
S2(k,m) = -(m+1) (m+1-k/2) (m+1-k) / 3k
For cubes,
Sn* n3 zn-1 =
(1+4z+z2 )/(1-z)4 if z¹1
or else z(-3) = 1/120.
S3
m = 0
m = 1
m = 2
m = 3
m = 4
m = 5
m = 6
k = 1
1/120
k = 2
-7/120
1/15
k = 3
-13/120
-13/120
9/40
k = 4
-7/240
-7/15
-7/240
8/15
k = 5
29/120
-91/120
-91/120
29/120
25/24
k = 6
91/120
-13/15
-63/40
-13/15
91/120
9/5
k = 7
1321/840
-599/840
-1919/840
-1919/840
-599/840
1321/840
343/120
S3(k,m) = k3 / 120 - (m+1)2 (m+1-k)2 / 4k
On the diagonal m = k-1 the decimated sum is indeed k3/120 as expected
from the theorem heralded above, which is the subject of the next section.
Next, we consider the series of biquadrates (or fourth powers ).
Sn* n4 zn-1 is
0 when z=1 and
(1+11z+11z2+z3 )/(1-z)5 when z¹1.
An infinite family of invariants, indexed by k,
is formed by the sums of the
k-decimations of a given series.
Those form again a series whose invariants are second-order invariants of the original series,
starting with the sum of the new series. This decimation procedure may be
carried on with the new series, and so forth...
(2018-05-20)
Divergent Fourier expansions
Harmonic analysis is a source of summable divergent series.
The Fourier expansion of the tangent function
is a summable divergent series. So is the expansion of its derivative:
½ tan (x/2) = sin x - sin 2x
+ ... + (-1)n+1 sin nx ...
¼ cos-2 (x/2) = cos x - 2 cos 2x
+ ... + (-1)n+1 n cos nx ...
Don't even think about trying the latter with x = p.
However, with x = 0, that's the sum of a series we're already
familiar with.
With x = p/2, we get:
(2019-08-09)
Asymptotic Solutions to Differential Equations
Definite differential properties of some divergent series.
Several methods exist to solve an
ordinary differential equation
as an asymptotic series about an hypothetical zero value of a nonzero parameter.
The WKB method is one,
but there are many others.
In particular, Carl Bender has pointed out that the usual way to solve a
perturbed Schrödinger equation expresses
the solution as a divergent asymptotic series.
Conversely, if we know a priori,
the solutions corresponding to the values 0 and 1 of a parameter,
such methods may yield a definite sum at point 1
for the ensuing asymptotic series about 0. Convergent or not.
Nicholas Mercator
(né Niklaus Kauffman, 1620-1687)
was the first to publish the series, in his Logarithmotechnia treatise (1668).
Isaac Newton (1643-1727) also discussed the series, which is
now called either the Mercator series or the Newton-Mercator series.
The Mercator series converges absolutely for |z| < 1.
For z = -1, it's an alternating series that converges to
-ln 2, as shown in 1650 by
Pietro Mengoli
(1626-1686).
For z = 1, we obtain the harmonic series
which was first shown to be divergent
by Nicolas Oresme (c.1350).
(2018-07-01) Decimating the Harmonic Series
Dealing with a multivalued analytic continuation.
f (1) is x
(a constant we'll identify later).
For z¹1,
we have essentially :
f (z) = - Log (1-z) / z
That's treacherous because Log is a
multivalued complex function
(defined modulo 2pi ) and we
must use a single analytic branch covering continuously
the closed unit disk, with the possible exception of z=1.
It turns out that all computer algebra systems (CAS) make the choice we need
by implementing their Log function with a
branch cut (or cliff)
on the negative half of the real axis.
For the sake of enlightenment, let's turn the auto-pilot off:
Consider a point z ¹ 1 on the unit circle:
z = eiq with 0 < q < 2p
1-z
=
1 - cos q - i sin q
=
2 sin2 q/2 - 2 i sin q/2 cos q/2
=
-2i sin q/2 eiq/2
=
2 sin q/2 ei (q-p)/2
In the above restricted range of q,
sin q/2 is positive. Therefore:
Log (1-z) = ln (2 sin q/2) + i (q-p)/2
+ i 2pn (for some integer n)
With n = 0, we get an analytic expression of f (z) valid on
the unit circle punctured at z=1,
consistent with f (0) = 1.
The decimation formula yields:
1
k
[
x -
k-1
exp (- 2pi (m+1) q/k )
{ ln (2 sin pq/k) + ip (q/k - ½) }
Quick numerical verification, with arbitrary precision :
One easy way to validate numerically the result of the transfinite
summation of a divergent series is to express that series as a sum of
other series which are either convergent series
(whose sums are computed numerically with or without acceleration procedures)
or divergent series of known sums.
(Conveniently, the series of the k-th powers of the positive integers
has zero sum when k is any positive even integer.)
To deal with the above case of the harmonic series,
we'll use countable additivity
to form a series with only three nonzero terms.
Our starting point is provided by one of the formal series which define
Bernoulli numbers (divided by n2,
for good measure):
1
=
1
-
1
+
1
+ 0 -
n2
+ 0 + ...
Bk nk-2
+ ...
n (en-1)
n2
2n
12
720
k!
Sn
1
=
p2
-
x
-
1
+ 0 +
0
+ 0 + ...
0
+ ...
n (en-1)
6
2
24
As the series on the left-hand-side converges nicely, its
sum S is easily obtained numerically (0.6843...) to arbitrary precision,
which yields:
x = p2 / 3
- 1 / 12 - 2 S
= 1.8378770664...
= Log 2p
Truthfully, the above verification is actually how
the transfinite sum of the harmonic function was first discovered
(on 2018-07-12) using a 12-place
decimal value (quickly obtained on a handheld calculator)
which Simon Plouffe's Inverter
identified immediately. A memorable magic moment !
At that time, I wasn't entirely sure that the postulate of
countable additivity was legitimate,
but that simple result gave me a great boost of confidence...
Arguably, x is the correct value to assign
to z at the simple pole 1.
The above goes against the popular guess that z(1)
ought to be defined as the Cauchy mean
of z about 1
(the constant term of the Laurent expansion):
(2012-07-31)
Summations of p-adic Integers
Any p-adic series whose
nth term is a multiple of n! converges !
The following series is clearly convergent
in p-adic integers for any p :
1 + 1 + 2 + 6 + 24 +
120 + ... + n! + ...
That's because the result of the sum modulo pk
is not influenced at all by the terms beyond a certain index m
(namely, the least integer whose factorial is a multiple of pk ).
This is also true if the radix (p) is not prime.
The decadic sum is
...4804323593105039052556442336528920420920314
The 2-adic sum is
...101110010110111111000011111011111101000011010
That series converges in p-adic integers for any radix p (prime or not)
and the sum is not invertible for some of them, which may be perceived as
its finite "factors". Those are the divisors of the following product:
Well before the more general notion of distributions was devised
(in 1944, by my late teacher Laurent Schwartz)
the Dutch mathematician
Thomas Stieltjes considered measures as generalized
derivatives of functions of bounded variations of a real variable.
Such functions are differences of two monotonous bounded functions;
they need not be differentiable or continuous.
(Stieltjes got his doctorate in Paris,
under Hermite and Darboux.)
Videos of Carl Bender at the PI (2011):
9:56
|
1:15:55
(2012-07-30)
Shanks Transformation (1955 & R.J. Schmidt 1941)
The transform of a sequence has the same limit but better convergence.
Motivation :
In a convergent sequence of the form
An = L + u vn
we may extract the limit L from 3 consecutive terms, by eliminating
u and v as follows:
An-1 = L + u vn-1
An = L + u vn
An+1 = L + u vn+1
So,
v = ( An - L )
/ ( An-1 - L )
= ( An+1 - L )
/ ( An - L )
Therefore,
( An - L ) 2
= ( An-1 - L )
( An+1 - L )
which implies :
L =
( An-1 An+1 - An2 )
/ ( An-1 + An+1 - 2 An )
Thus, the right-hand-side of that expression forms a sequence whose terms are
expected to be close to the limit of An
even when An is not of the special form quoted above.
This motivates the following introduction of a new sequence Sn ,
which is defined for positive indices whenever the denominator doesn't vanish:
Shanks transform Sn
of the sequence An
Sn =
An-1 An+1 - An2
An-1 + An+1 - 2 An
We observe that Shanks' transformation commutes with translation :
Thus, wlg, we may focus on the analysis of sequences whose limit is zero.
(The difference between a convergent sequence and its limit is of this type.)
An
Shanks Transform of An
vn
0
1 / n
1 / 2n
1 / np
~ 1 / (p+1)np
(-1)n / n
(-1)n+1 / [ 4n (n2 - ½) ]
(-1)n / np
~ (-1)n+1 p / [ 4 np+2 ]
The table shows that the convergence of a sequence that alternates above and below its limit
is greatly accelerated by Shanks' transformation (the distance to the
limit is essentially divided by the square of the index n).
Shanks's transformation is thus highly recommended for alternating series.
No such luck when the sequence approaches its limit from one side only.
The Shanks transform then offers only marginal improvement (by dividing
the distance to the limit by a constant factor, which is usually 2 or 3).
In that case, the approach described in the next section
is preferred.
(2012-07-30)
Richardson Extrapolation
(1911 &
Takebe 1722)
Accelerating the convergence of
An = L + k1 / n + k2 / n2 + ...
This (very common) pattern of convergence is
the case where the above transformation of Shanks
has the poorest performance. By optimizing for this pattern, we'll
provide a convergence improvement in cases where the Shanks transformation does not deliver.
The method is similar, we eliminate 2 parameters between 3 equations:
An-1 (n-1)2 = L (n-1)2
+ k1 (n-1) + k2
An n 2 = L n 2
+ k1 n
+ k2
An+1 (n+1)2 = L (n+1)2
+ k1 (n+1) + k2
Subtract twice the second equation from the sum of the other two:
An-1 (n-1)2
- 2
An n 2
+
An+1 (n+1)2
= 2 L
This motivates the definition of the (order 2)
Richardson transformation:
Richardson transform Rn
of the sequence An
Rn =
(n-1)2 An-1
- 2 n2 An +
(n+1)2 An+1
2
Richardson's transform is a linear map
that commutes with translation.
So, without loss of generality we can restrict the analysis of its performance
to convergent sequences whose limit is zero (consider such a sequence
as the difference between some other sequence and its limit, if you must).
An
Richardson Transform of An
~
1 / n
0
1 / n 2
0
1 / n 3
1 / n(n2-1)
1 / n 3
1 / n 4
(3n2-1) /
n2(n2-1)2
3 / n 4
1 / n 5
(6n4-3n2+1) /
n3(n2-1)3
6 / n 5
1 / n 6
(10n6-5n4+4n2-1) /
n4(n2-1)4
10 / n 6
1 / n 7
(15n8-5n6+10n4-5n2+1) /
n5(n2-1)5
15 / n 7
(-1)n / n
2n (-1)n+1
(-1)n / n 2
2 (-1)n+1
Thus, unlike the Shanks transform, Richardson's transformation
is absolutely catastrophic when applied to the partial sums of an alternating series.
For a typical nonalternating series, it does a perfect job at the cancellation
of the leading terms it's designed to handle and leaves the next order of magnitude virtually untouched.
However, the bad influence of higher-order error terms is significantly amplified (possibly fatally so).
Takebe (or "Tatebe")
=
Takebe Katahiro
=
Takebe Kenko (1664-1739)
"Our" Takebe was the younger brother of Takebe Kataaki (1661-1716) also a
student of Seki.
(2012-09-29)
On a new acceleration method
(Michon, 2012)
Accelerating the convergence of
An = L + k / (n-a) + ...
The transformations presented in the previous section are somewhat unsatisfactory
because they involve explicitly the particular indexation of the sequence
(the value of n). Clearly, if we tune a convergence acceleration to
a truncated expansion of the shape presented here, the index n won't be involved
because the presence of the parameter a
makes the optimal result invariant by translation of the index.
Note that, if the correction terms of order 1/n3 and beyond are neglected,
our new target is of the same magnitude as our previous one, with
k1 = k and k2 = ak.
Again, we eliminate 2 parameters between 3 equations:
An-1 (n - a - 1)
= L (n - a - 1) + k
An (n - a)
= L (n - a) + k
An+1 (n - a + 1)
= L (n - a + 1) + k
Subtract twice the second equation from the sum of the other two:
( An-1
+ An+1 - 2 An )
(n - a)
+
( An+1 -
An-1 )
= 0
Let's also subtract the second equation from the third:
( An+1 -
An )
(n - a)
+ An+1 = L
Eliminating (n - a)
between those two equations, we obtain:
L =
[ 2 An-1 An+1
- An (
An-1
+ An+1 ) ]
/ ( An-1
+ An+1 - 2 An )
This motivates the following definition of a new sequence Bn ,
which is valid for positive indices whenever the denominator doesn't vanish:
Transform Bn
of the sequence An
Bn =
2 An-1 An+1
- An (
An-1
+ An+1 )
An-1 + An+1 - 2 An
You may want to note that the Shanks transform of
An is (An+Bn)/2.
As this new transform commutes with translation
(the reader is encouraged to check that directly) we may study its performance, without
loss of generality, for sequences whose limits are zero:
An
Bn =
Bn ~
1 / n
0
1 / n 2
-1 / (3n2-1)
-1 / 3n 2
1 / n 3
-3n / (6n4-3n2+1)
-1 / 2n 3
1 / n 4
-(6n2+1) /
(10n6-5n4+4n2-1)
-3 / 5n 4
(-1)n / n
-2n (-1)n/ (2n2-1)
- (-1)n/ n
(-1)n / n 2
-(2n2+1) (-1)n/ (2n4-n2-1)
- (-1)n/ n2
vn
-vn
Thus, the above transform is very effective when the leading
error term is harmonic (1/n).
For other types of convergence, the above table suggest using
a linear mix of A and B for best acceleration,
as investigated next.
(2012-09-30)
Universal Convergence Acceleration
Accelerating all typical analytical sequences.
Building on the above,
let's introduce a parameter u and define:
A'n =
[ (1-u) An + (1+u) Bn ] / 2
This way, the original sequence is obtained for u = -1,
its Shanks transform for u = 0 and the sequence
B of the previous section for u = 1.
A'n =
(1+u) An-1 An+1
- u An (
An-1
+ An+1 ) - (1-u) An2
An-1 + An+1 - 2 An
=
An-1 An+1
- An2
- u
(An - An-1 )
(An+1 - An )
An-1 + An+1 - 2 An
Or, equivalently:
Parametrized transformation of the sequence An
A'n =
An + (1+u)
(An - An-1 )
(An+1 - An )
(An - An-1 )
-
(An+1 - An )
The invariance by translation of this parametrized transform
allows us to study it only for sequences whose limit is zero
(without loss of generality among convergent sequences).
Of course, we'll seek the value of u which provides the best acceleration of convergence.
To use the example already analyzed, if An = L+k/n then
A'n = L + (1-u) k/n
As we already know, the best value of u
is indeed +1 (which yields a constant sequence equal to
the limit L).
Here are a few other cases:
Optimal values of u
An
u
A'n ~
1 / n
1
0
1 / n 2
1/2
-1 / 12n 4
1 / n 3
1/3
-1 / 6n 5
1 / n 4
1/4
-1 / 4n 6
1 / n 5
1/5
-1 / 3n 7
1 / n 6
1/6
-5 / 12n 8
1 / n p
1/p
(1-p) / 12n p+2
(-1)n/ n
0
-(-1)n/ 4n 3
For the partial sums of alternating series, the Shanks transform
(u=0) is optimal. Otherwise,
we can typically build an optimized sequence as:
An ,
A'n ,
A''n ,
A'''n ,
A''''n ,
A'''''n ,
...
For this, we use a special sequence of different parameters
determined by the expected way the sequence approaches its limit
asymptotically.
Typically (but not always) the original sequence
approaches its limit with a remainder roughly proportional to
1/n and one order of magnitude is gained with each iteration
using the sequence:
The computation is particularly easy to perform using a spreadsheet calculator.
We illustrate this by the following computation to 6 decimal places
of the
sum of the reciprocals of the squares,
based on the first 7 terms in the series
(9 terms are given to show that the last two are useless).
Highlighted in blue are the Shanks transforms of the extreme diagonals.
z(2) =
p2/ 6
= 1.644934066848226436472415...
n
An
u0 = 1
u1 = 1/2
u2 = 1/3
u3 = 1/4
1
1.000000
1.644704
2
1.250000
1.650000
1.644921
3
1.361111
1.646825
1.644661
1.644934
4
1.423616
1.645833
1.644811
1.644934
5
1.463611
1.645429
1.644868
1.644934
1.644934
6
1.491389
1.645235
1.644895
1.644934
7
1.511797
1.645130
1.644909
1.644934
8
1.527422
1.645069
1.644931
9
1.539768
1.644909
Although many terms of the basic sequence would be easy to compute in this
didactic example, the method is meant to handle situations where this is not the case.
In theoretical physics (quantum field theory)
and pure mathematics, we may only have a few terms available and only
a fuzzy understanding of the behavior of the sequence whose limit has to be guessed
as accurately as possible.
Incidentally, with standard limited-precision floating-point arithmetic,
the relevant computations presented above will be very poorly approximated
because we keep subtracting nearly-equal quantities.
As a rule of thumb, about half the available precision is wasted.
A 13-digit spreadsheet is barely good enough to reproduce the above
6½-digit table.
Extensions of it would be dominated by the glitches caused by limited precision.
Such pathological behavior is lessened by the approach described next.
(2012-10-02) Accelerating a series by transforming its terms.
The series counterpart of the parametrized transform for sequences.
If the sequence An is the partial sum of the series of term
an , then we have
an = An - An-1
(for n≥1)
and the above boils down to:
A'n =
An + (1+u)
anan+1
an - an+1
Subtracting from that value the counterpart for A'n-1 , we obtain:
a'n =
an + (1+u)
anan+1
- (1+u)
an-1an
an - an+1
an-1 - an
(2012-09-30) The next order...
An aborted attempt.
Let's target sequences of the form
An = L
+ k1 / (n-a)
+ k2 / (n-a)2
For the purpose of the following computation, we get rid of indices by
considering four consecutive terms in the sequence (A,B,C,D)
and introducing the quantity x
that differs from (n-a) by some integer.
We seek an expression of the limit L as a function of A,B,C,D
by eliminating the three quantities x,
k1 and k2 between the following
four equations:
A (x+0) 2 = L (x+0) 2 +
k1 (x+0) + k2
[ 1] [ 1]
B (x+1) 2 = L (x+1) 2 +
k1 (x+1) + k2
[-3] [-2]
C (x+2) 2 = L (x+2) 2 +
k1 (x+2) + k2
[ 3] [ 1]
D (x+3) 2 = L (x+3) 2 +
k1 (x+3) + k2
[-1] [ 0]
The two columns of coefficients yield respectively these combinations:
( A - 3B + 3C - D ) x2 + ( -6B + 12C -6D ) x + ( -3B + 12C - 9D) = 0
( A - 2B + C ) x2 + ( -4B + 4C ) x + ( -2B + 2C ) = 2L x
Eliminating x between those two quadratic equations yields:
A-3B+3C-D A-2B+C
-3B+12C-9D -2B+2C
2
=
A-3B+3C-D A-2B+C
-6B+12C-6D -2B+4C-2L
.
-6B+12C-6D -2B+4C-2L
-3B+12C-9D -2B+2C
Unfortunately, this is now a quadratic equation in L.
(2018-05-15) Summation by Factorial Series
A forgotten method introduced by James Stirling in 1730.
(2019-11-21) Physical meaning for the summation of divergent series
Why divergent series may be unavoidable in quantum physics.
In the tamest type of infinite series, the
absolutely convergent series,
all the familiar propertie of finite sums are preserved,
including commutativity and associativity.
This comfortable fact may be useful for numerical applications
but it's simply roo restrictine for the way our quantum world seems to be constructed.
A certain historical order is essential to the way infinitely many
probability amplitudes are combined, yet trajectories are not defined.
There's no such thing as a partial history. Only the whole can be considered.
The whole sum need not be the limit of partial sums.
Fractional derivatives are part of mathematical folklore.
They were first defined in 1695 by Leibniz in a letter
to Guillaume de l'Hospital.
They apply in any context where fractional exponents are defined.
They can be motivated by the following expression (easily established by induction on k)
for the continuous linear operator Dk defined as k repetions of the
ordinary derivation operator D = D1 when k is a positive integer
(this expression is consistent with the convention that D0 is the identity operator)
when it's applied to a monomial. By linearity, that definition extends to all polynomials and, by continuity,
to all analytic functions.
Dk xm = [ m! / (m-k)! ] xm-k
Using the Gamma function, we obtain an equivalent expression which remains valid when k isn't an integer.
Dk xm = [ G(m+1) / G(m-k+1) ] xm-k
That expression holds for almost all values of k, even complex ones
(exceptions arise because G is undefined for negative odd integers).
In this, x is understood to be a positive real number
(otherwise, raising x to a fractional power is problematic).
To remove that restriction, we may consider instead the compensated operator which is
defined in any power-associative algebra, including rings of matrices:
Jk xm = [ G(m+1) / G(m-k+1) ] xm
Jk doesn't change the degree of a polynomial, unless said degree is less than k.
(2018-06-23) The Cauchy product of two series.
