Convergence improvement, also called convergence acceleration or accelerated convergence, transforms a series or sequence so that it approaches its limit more rapidly. Convergence improvement can be effected by forming a linear combination with a series whose sum is known. Useful sums include
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(1)
|
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(2)
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(3)
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(4)
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For a positive integer 
, the last formula follows by telescoping.
Kummer's transformation takes a convergent series
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(5)
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and another convergent series
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(6)
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with known 
,
with 
for all sufficiently large 
, such that
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(7)
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Then the same sum can be written as
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(8)
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(Abramowitz and Stegun 1972). The residual terms satisfy 
, which can improve convergence.
The Euler transform rewrites a convergent series of the form
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(9)
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as
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(10)
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provided the transformed series converges, where 
is the forward difference operator and
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(11)
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(Abramowitz and Stegun 1972, Beeler et al. 1972). This transformation often accelerates slowly convergent alternating series.
A general technique for accelerating a series is to expand the summand in powers of the reciprocal of its index and interchange the order of summation when the interchange is justified. A symbolic form for this expansion can sometimes allow the original index to be summed symbolically. For example, consider the case of the sum
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(12)
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(OEIS A085361) that arises in the definition of the Alladi-Grinstead constant. The summand can be expanded about infinity to get
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(13)
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Since all terms in the double series are nonnegative, interchanging the order of summation gives
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(14)
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where 
is the Riemann zeta function, which converges
much more rapidly.
Aitken's delta-squared process uses the transformation
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(15)
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where 
is the 
th
partial sum of a sequence 
.
For 
and 
,
the transformation can accelerate convergence
(Hamming 1986, p. 205). In particular, 
can be written
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(16)
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The application of this transformation can be efficiently carried out using Wynn's epsilon method. Set 
, 
, and use
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(17)
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for 
,
1, ..., wherever the denominator is nonzero. For a positive
integer 
,
the even entries 
give Shanks'
transformation of order 
, based on 
, 
, 
. In particular, 
. Higher orders generally differ from
repeated applications of Aitken's delta-squared
process.
Given a series of the form
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(18)
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where 
is analytic in a neighborhood of the closed
unit disk, and
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(19)
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then the series can be rearranged to
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(20)
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where
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(21)
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is the Maclaurin series of 
and 
is the Riemann zeta
function (Flajolet and Vardi 1996). The transformed series
exhibits geometric convergence. Similarly, if 
is analytic in a neighborhood
of 
for some positive integer 
and still satisfies 
at 0, then
![S=sum_(n=1)^(n_0-1)f(1/n)+sum_(m=2)^inftyf_m[zeta(m)-1/(1^m)-...-1/((n_0-1)^m)],](/images/equations/ConvergenceImprovement/NumberedEquation14.svg) |
(22)
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which converges geometrically (Flajolet and Vardi 1996). Equation (22) can also be used to further accelerate the convergence of series (20).
