Research Article
Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series
Joseph Gaskin*
Issue:
Volume 10, Issue 3, June 2024
Pages:
33-37
Received:
3 September 2024
Accepted:
19 September 2024
Published:
10 October 2024
DOI:
10.11648/j.ijtam.20241003.11
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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 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.
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 te...
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