Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Joseph Gaskin

doi:doi:10.11648/j.ijtam.20241003.11

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Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Joseph Gaskin^*

Published in International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)

Received: 3 September 2024 Accepted: 19 September 2024 Published: 10 October 2024

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Abstract

Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the T_{φ, ρ}transformations on the partial sums of any convergent series. Furthermore, I prove that these T_{φ, ρ}transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)
DOI	10.11648/j.ijtam.20241003.11
Page(s)	33-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Series, Accelerators, Logarithmic, Convergence

1. Introduction

This paper is devoted to accelerating the convergence of an infinite series

\sum_{n = 1}^{\infty} a_{n} = S < \infty,

that satisfies

\lim_{n \to \infty} |\frac{{S - a}_{n + 1}}{S - a_{n}}| = 1

. We call such series logarithmic series. Note that if

a_{n + 1} \geq a_{n}

and if

\lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} = 1

, then the convergent series

\sum_{n = 1}^{\infty} a_{n}

, is a logarithmic series

[2, 10]

Throughout this paper,

f (x) = a_{x}

shall denote a continuous, positive valued, and a monotonically decreasing function on

[1, \infty)

, that satisfies:

\lim_{x \to \infty} \frac{f (x + 1)}{f (x)} = 1

and

\int_{1}^{\infty} f (x) dx < \infty

μ

is a positive real number greater than 1, then we extend the traditional definition of the partial sum of an infinite series

[15]

to define the

μ^{th}

partial sum,

S (μ)

, of

\sum_{n = 1}^{\infty} a_{n}

as follows:

S (μ) = a_{1} + a_{2} + \dots a_{[μ]} + (μ - [μ]) a_{[μ] + 1}

, where

[μ]

is the greatest integer less than

μ

{\{a_{n}\}}_{n = 1}^{\infty}

and

{\{b_{n}\}}_{n = 1}^{\infty}

are sequences converging to A and B respectively, such that

\lim_{n \to \infty} |\frac{a_{n} - A}{b_{n} - B}| = 0

, then we say that

{\{a_{n}\}}_{n = 1}^{\infty}

converges more rapidly than

{\{b_{n}\}}_{n = 1}^{\infty}

[11]

, S. Mukherjee et. al cite a well-known theorem by J.P. Delahaye and Germain-Bonne

[5]

which states that: it is impossible to construct a series accelerator which can accelerate the convergence of all convergent series.

In this paper, we show that, for a large group of convergent, logarithmic series,

\sum_{n = 1}^{\infty} a_{n}

, there exists a transformation,

T

, on the partial sums

[6, 7]

of the series and an increasing function

φ

[1, \infty)

, such that

φ (n) > n

T (S (φ (n)))

\to S

\lim_{n \to \infty} |\frac{T (S (φ)) - S}{S (φ (n)) - S}| = 0

[1]

(see pages 11 and 12), Brezinski shows that if:

T_{φ, ρ} (S (n)) = S (φ (n)) + D_{φ (n)}

, then

\lim_{n \to \infty} (\frac{D_{φ (n)}}{S - S (φ (n))}) = 0

is equivalent to

\lim_{n \to \infty} |\frac{T_{φ, ρ} (S (n)) - S}{S (φ (n)) - S}| = 0

D_{φ (n)}

is then called a perfect estimation of the error of

S (φ (n))

Thus, we shall prove that if

{\{a_{n}\}}_{n = 1}^{\infty}

is a sequence of decreasing positive terms such that:

\lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} = 1

and

0 < \lim_{n \to \infty} (\frac{S (φ (n)) - S (φ (n - 1))}{a_{n}}) = ρ < 1

then

D_{φ (n)} = (\frac{S (n) - S (φ (n))}{1 - \frac{1}{ρ}})

will be a perfect estimation of the error of

S (φ (n))

This result leads naturally to a class of series accelerators:

T_{φ, ρ} (S (n)) = S (φ (n)) + D_{φ (n)} = S (φ (n)) + (\frac{S (n) - S (φ (n))}{1 - \frac{1}{ρ}})

The

T_{φ, ρ}

transformations are extensions of the

T_{+ m}

[3]

accelerators by Clark and Gray (see

[4]

) since

ρ

works for all real numbers belonging to the interval

(0, 1)

Furthermore, we shall also establish an interesting relationship between

ρ

, a convergent series

\sum_{n = 1}^{\infty} a_{n}

, and an equation:

φ^{'} (x) f (φ (x)) = ρf (x)

It is worthy to note that if

φ (n) = n^{2}

, then the transformation:

T_{φ, ρ} (S (n)) = (\frac{S (n) - \frac{1}{ρ} S (φ (n))}{1 - \frac{1}{ρ}})

can be used to accelerate the convergence of the series

\sum_{n = 1}^{\infty} \frac{1}{n {(\ln (n))}^{2}}

. I am not aware of any other series accelerator capable of accelerating the convergence of this series.

2. Section 1

We shall begin this section with an interesting result which characterizes all decreasing sequences

{\{a_{n}\}}_{n = 1}^{\infty}

, whose sum,

\sum_{n = 1}^{\infty} a_{n}

, converges.

Theorem 1

Let

f

be a continuous, positive valued, and decreasing function on

[1, \infty)

. Then, for each

ρ \in (0, 1)

\sum_{n = 1}^{\infty} f (n)

converges if and only if there exists a differentiable function

φ

[1, \infty)

, such that:

φ (x) > x

, and

φ^{'} (x) f (φ (x)) = ρf (x)

Proof:

Suppose that

\sum_{n = 1}^{\infty} f (n)

converges. Then,

\int_{1}^{\infty} f (t) dt

converges

[8, 14]

. Furthermore, there exists

F (x)

, a negative-valued antiderivative of

f (x)

that satisfies

\lim_{x \to \infty} F (x) = 0

[9]

. Let

φ (x) = F^{- 1} (ρF (x))

Differentiating

φ (x)

with respect to

x

, we get

φ^{'} (x) f (φ (x)) = ρf (x)

Since

F (x)

is increasing on

[1, \infty)

ρF (x) > F (x)

Thus,

φ (x) = F^{- 1} (ρF (x))

F^{- 1} ((F (x)) = x

Next, suppose that

φ (x) > x

, and that

φ^{'} (x) f (φ (x)) = ρf (x)

Then,

\frac{φ^{'} (x) f (φ (x))}{f (x)} = ρ < 1 .

Therefore, by

[2]

(see page 44),

\sum_{n = 1}^{\infty} f (n)

converges.

Before proving our next Theorem, we shall give a couple of definitions. These definitions will mainly serve the purpose of simplifying the statement of Theorem 2.

Definition 1

φ

is a function satisfying the following conditions:

φ (x) > x

and

φ^{'} (x) \geq 1

, and if

\lim_{x \to \infty} \frac{φ^{'} (ξ_{x})}{φ^{'} (η_{x})} = 1

, where

ξ_{x} and η_{x} \in [x, x + 1]

then we say that

φ

is logarithmic on

[1, \infty)

Definition 2

Let

φ (x)

be a positive valued function and let

ρ

be an arbitrarily chosen positive number lying in the interval

(0, 1)

. We define the

T_{φ, ρ}

transformations as follows:

T_{φ, ρ} (S (n)) = S (φ (n)) + (\frac{S (n) - S (φ (n))}{1 - \frac{1}{ρ}})

Theorem 2

Suppose that

\sum_{n = 1}^{\infty} f (n)

is a convergent, logarithmic series of a monotonically decreasing sequence of positive terms and that

φ

is logarithmic on

[1, \infty)

. Then,

T_{φ, ρ} (S (n)) = (\frac{S (n) - \frac{1}{ρ} S (φ (n))}{1 - \frac{1}{ρ}})

converges more rapidly than

S (φ (n))

Proof:

First, we shall show that

\lim_{n \to \infty} (\frac{S (φ (n)) - S (φ (n - 1))}{f (n)}) = ρ

To this end, note that

\frac{S (φ (n)) - S (φ (n - 1))}{f (n)} \leq \frac{(φ (n)) - φ (n - 1)) f (φ (n - 1))}{f (n)}

\leq \frac{φ^{'} (ξ_{n}) f (φ (n - 1))}{f (n)}

\leq (\frac{φ^{'} (n - 1) f (φ (n - 1))}{f (n - 1)}) (\frac{f (n - 1)}{f (n)}) (\frac{φ^{'} (ξ_{n})}{φ^{'} (n - 1)})

\leq ρ (\frac{f (n - 1)}{f (n)}) (\frac{φ^{'} (ξ_{n})}{φ^{'} (n - 1)})

, where

ξ_{n} \in [n - 1, n]

Thus,

\lim^{̅} \frac{S (φ (n)) - S (φ (n - 1))}{f (n)} \leq ρ

Similarly,

\frac{S (φ (n)) - S (φ (n - 1))}{f (n)} \geq \frac{(φ (n)) - φ (n - 1)) f (φ (n_)}{f (n)}

\geq (\frac{φ^{'} (η_{n}) f (φ (n))}{f (n)})

\geq (\frac{φ^{'} (n) f (φ (n))}{f (n)}) (\frac{φ^{'} (η_{n})}{φ^{'} (n)})

\geq ρ (\frac{φ^{'} (η_{n})}{φ^{'} (n)})

, where

η_{n} \in [n - 1, n]

Therefore,

\lim_{̲} \frac{S (φ (n)) - S (φ (n - 1))}{f (n)} \geq ρ .

Hence,

\lim_{n \to \infty} \frac{S (φ (n)) - S (φ (n - 1))}{f (n)} = ρ

[13]

Now,

S (φ (n)) - S (n))

and

S (φ (n)) - S

both converge to 0 monotonically.

Therefore, by

[2]

(see page 413) and above, we have that:

\lim_{n \to \infty} \frac{S (φ (n)) - S (n)}{S (φ (n)) - S} = 1 - \lim_{n \to \infty} (\frac{f (n)}{S (φ (n)) - S (φ (n - 1))}) = \frac{ρ - 1}{ρ}

Hence,

\lim_{n \to \infty} (\frac{T_{φ, ρ} (S (n)) - S}{S (φ (n)) - S})

= \lim_{n \to \infty} (\frac{T_{φ, ρ} (S (n)) - S (φ (n)) + S (φ (n)) - S}{S (φ (n)) - S})

1 - \lim_{n \to \infty} (\frac{D_{φ (n)}}{S (φ (n)) - S})

= 1 - (\frac{1}{1 - \frac{1}{ρ}}) \lim_{n \to \infty} (\frac{S (φ (n)) - S (n)}{S (φ (n)) - S})

= 1 - (\frac{1}{1 - \frac{1}{ρ}}) (1 - \frac{1}{ρ}) = 0

Thus,

D_{φ (n)} = (\frac{S (n) - S (φ (n))}{1 - \frac{1}{ρ}})

is a perfect estimation of the error of

S (φ (n))

, whenever

φ

is logarithmic on

[1, \infty)

, and

T_{φ, ρ} (S (n)) = S (φ (n)) + D_{φ (n)}

Corollary 2.1

Suppose that

\sum_{n = 1}^{\infty} f (n)

is a convergent logarithmic series of a monotonically decreasing sequence of positive terms. If

\lim_{x \to \infty} \frac{f (φ (x + 1))}{f (φ (x))} = 1

, then,

T_{φ, ρ} (S (n)) = (\frac{S (n) - \frac{1}{ρ} S (φ (n))}{1 - \frac{1}{ρ}})

converges more rapidly than

S (φ (n))

Proof:

Note that

\frac{φ^{'} (x + 1)}{φ^{'} (x)} = \frac{f (x + 1)}{f (x)} . \frac{f (φ (x + 1))}{f (φ (x))}

, and

\lim_{x \to \infty} \frac{f (x + 1)}{f (x)} = 1

Next, we provide some examples showing how to determine

φ (x)

for a few convergent logarithmic series.

3. Examples

Example 1. Let

f (x) = \frac{1}{x^{2}}

. Then,

F (x) = \frac{- 1}{x}

and therefore

F^{- 1} (x) = \frac{- 1}{x}

Hence,

φ (x) = \frac{x}{ρ}

. If

ρ = \frac{1}{m}

, where

m

is a positive integer, then the

T_{φ, ρ}

series accelerators reduce to the

T_{+ m}

transformations (see

[4]

, page 268, definition 4.1).

Example 2. Let

f (x) = \frac{1}{{x (\ln (x))}^{2}}

. Then,

F (x) = \frac{- 1}{lnx}

and

F^{- 1} (x) = e^{\frac{- 1}{x}}

. Thus,

φ (x) = x^{\frac{1}{ρ}}

. Note that

φ

is logarithmic on

[1, \infty)

Example 3. Let

f (x) = \frac{1}{xlnx {(\ln (lnx)))}^{2}}

. Then,

F (x) = \frac{- 1}{\ln (\ln (x)}

and

F^{- 1} (x) = e^{e^{\frac{- 1}{x}}}

. Hence,

φ (x) = e^{{(lnx)}^{\frac{1}{ρ}}}

and

φ

is logarithmic on

[1, \infty)

since

φ^{'} (x) = (\frac{{(lnx)}^{\frac{1 - ρ}{ρ}}}{ρx} φ (x))

Now,

\sum_{k = 1}^{\infty} \frac{1}{k^{2}} =

1.644.93 (5 decimal places)

[12]

In the following table,

S (n) = \sum_{k = 1}^{n} \frac{1}{k^{2}}

(5 decimal places). We observe that it will take more than the first 20,000 terms of the series to attain 5 decimal places of accuracy. However, if one applies the

T_{φ, ρ}

series accelerators, one only needs at most the first 400 terms to achieve the same level of accuracy.

Table 1. Partial sums of

\sum_{k = 1}^{\infty} \frac{1}{k^{2}}

$n$	$S (n)$	$S (2 n)$	$T_{2 n, . 5}$ ₌ $2 S (2 n) - S (n)$	$S (4 n)$	$T_{4 n, . 25}$ = $\frac{(4 S (n) - S (n))}{3}$
10	1.54977	1.59616	1.64256	1.62024	1.64374
100	1.63498	1.63995	1.64491	1.64244	1.64492
200	1.63995	1.64244	1.64493	1.64369	1.64493
1000	1.64393	1.64443	1.64493	1.64468	1.64493
10000	1.64483	1.64488	1.64493	1.64491	1.64493
100000	1.64492	1.64493	1.64493	1.64493	1.64493

4. Conclusion

In conclusion, we have shown if

\sum_{n = 1}^{\infty} a_{n}

is a convergent logarithmic series and if the

φ

function associated with this series is logarithmic, then the convergence of

\sum_{n = 1}^{\infty} a_{n}

can be accelerated by the

T_{φ, n}

series accelerators.

Consequently, any convergent logarithmic series that can be written as a linear combination of logarithmically convergent series such that the associated

φ

functions converge logarithmically, can be accelerated by the

T_{φ, ρ}

series accelerators.

Acknowledgments

I wish to express my heartfelt thanks to Dr. William Ford for the useful insights he provided me in the field of series acceleration.

Author Contributions

Joseph Gaskin is the sole author. The author read and approved the final manuscript.

Conflicts of Interest

The author declares no conflict of interest.

References

[1]	Brezinski, C., and Zaglia, R., Extrapolation Methods, Theory and Practice, Studies in Computational Mathematics 2, Elsevier, 2013.
[2]	Bromwich, T. J., An Introduction to the Theory of Infinite Series, Alpha Editions, www.alphaedis.com (2020).
[3]	Belghaba K., On the Transformation T_+m due to Gray and Clark, Journal of Mathematics and Statistics, Vol. 3, https://doi.org/10.3844/jmssp,2007.243.248
[4]	H. L. Gray, and W. D. Clark, “On a Class of Nonlinear Transformation and their Applications to the Evaluation of Infinite Series,” Journal of Research of the National Bureau of Standards-B. Mathematical Sciences, Vol. 73B, No. 3, July-September 1969.
[5]	J. P. Delahaye and B. Germain-Bonne, SIAM Journal on Numerical Analysis, Vol. 19, No. 4, 1982, pp 840-844.
[6]	C. Brezinski, Convergence acceleration during the 20^th century, Journal of Computational and Applied Mathematics, Volume 122, Issues 1-2, October 2000, pp 1-2.
[7]	d’AsperemontA, Scieur D, Taylor A2021, Acceleration Methods, Foundations and Trends (R) in Optimization, 5(1-2), pp1-245 nowpublishers.com
[8]	Briggs, Cochran, Gillet, Shultz, Calculus Early Transcendentals, 3^rd Edition, ISBN13: 978-0-13-476684-3, 2019, pp 674.
[9]	Howard Anton, I. C. Bivens, Stephen Davis, Calculus, 12^th Edition, 2021, ISBN-13:978-1119778127, ISBN-10:1119778123, pp 290.
[10]	Andrew H. Van Tuyl, Acceleration of Convergence of a family of logarithmically convergent sequences, Mathematics of Computation, Vol 63, Number 207, July 1994, pp 229.
[11]	S. Mukherjee, B-S Hua, N. Umetani, D. Meister, Neural sequence transformation, Computer Graphics Forum, The Eurographics Association and John Wiley & Sons Ltd, 2021, pp 132.
[12]	Khan SA, International Journal of Applied Mathematics, ISBN: 314-8060, Volume 33, 2020. https://doi.org/10.1273/ijam.v33i2.6
[13]	Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4^th Edition, ISBN-13: 978-0471433316, 2011, pp 400-418.
[14]	James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, ISBN-13: 978-1337624183, 9^th Edition, April 2020, Chapter 7.8, pp 534.
[15]	Britannica, The Editors Encyclopedia, “Infinite series”, Encyclopedia Britannica, August 2024.

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APA Style

Gaskin, J. (2024). Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. International Journal of Theoretical and Applied Mathematics, 10(3), 33-37. https://doi.org/10.11648/j.ijtam.20241003.11

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ACS Style

Gaskin, J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int. J. Theor. Appl. Math. 2024, 10(3), 33-37. doi: 10.11648/j.ijtam.20241003.11

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AMA Style

Gaskin J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int J Theor Appl Math. 2024;10(3):33-37. doi: 10.11648/j.ijtam.20241003.11

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@article{10.11648/j.ijtam.20241003.11,
  author = {Joseph Gaskin},
  title = {Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {33-37},
  doi = {10.11648/j.ijtam.20241003.11},
  url = {https://doi.org/10.11648/j.ijtam.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.11},
  abstract = {Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.
},
 year = {2024}
}

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TY  - JOUR
T1  - Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

AU  - Joseph Gaskin
Y1  - 2024/10/10
PY  - 2024
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DO  - 10.11648/j.ijtam.20241003.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 33
EP  - 37
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.11
AB  - Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

VL  - 10
IS  - 3
ER  -

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Author Information

Joseph Gaskin

Department of Mathematics and Physics, Southern University and A&M College, Baton Rouge, USA

Research Fields: Real Analysis, Sequences and series, Number Theory, Differential equations, Point Set Topology

Contact Email

http://orcid.org/0009-0004-2781-647X

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Table 1

Table 1. Partial sums of $\sum_{k = 1}^{\infty} \frac{1}{k^{2}}$ .

Plain Text BibTeX RIS

APA Style

Gaskin, J. (2024). Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. International Journal of Theoretical and Applied Mathematics, 10(3), 33-37. https://doi.org/10.11648/j.ijtam.20241003.11

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Gaskin, J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int. J. Theor. Appl. Math. 2024, 10(3), 33-37. doi: 10.11648/j.ijtam.20241003.11

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AMA Style

Gaskin J. Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series. Int J Theor Appl Math. 2024;10(3):33-37. doi: 10.11648/j.ijtam.20241003.11

Copy | Download

@article{10.11648/j.ijtam.20241003.11,
  author = {Joseph Gaskin},
  title = {Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {33-37},
  doi = {10.11648/j.ijtam.20241003.11},
  url = {https://doi.org/10.11648/j.ijtam.20241003.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.11},
  abstract = {Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

AU  - Joseph Gaskin
Y1  - 2024/10/10
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241003.11
DO  - 10.11648/j.ijtam.20241003.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 33
EP  - 37
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.11
AB  - Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number ρ and a differentiable function φ. Then, I use ρ and φ to define the Tφ, ρ transformations on the partial sums of any convergent series. Furthermore, I prove that these Tφ, ρ transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute φ if one is given the convergent series of decreasing, positive terms.

VL  - 10
IS  - 3
ER  -

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