Recycling Stirling's series acceleration technique

2017 ◽  
Vol 101 (550) ◽  
pp. 69-82
Author(s):  
Paul Levrie ◽  
Amrik Singh Nimbran
Keyword(s):  
Rate Of Convergence ◽  
Infinite Series ◽  
Mathematical Analysis ◽  
Special Form ◽  
Partial Sums ◽  
Repeated Application ◽  
Euler’S Method ◽  
Acceleration Technique ◽  
General Series ◽  
Series Acceleration

Infinite series are an important topic in mathematical analysis and convergence is the most crucial concept in the theory of infinite series. The speed at which the sequence of the partial sums of a series approaches its limiting sum has been a subject of investigation for many a mathematician. Euler [1], Kummer [2] and Markoff [3] all developed techniques for accelerating the convergence of slowly converging series. Euler's method only works for alternating series, Kummer's and Markoff's are suitable for general series with a special form.Gosper [4] illustrates how the rate of convergence of infinite series can be accelerated by a suitable splitting of each term into two parts and then combining the second part of the n th term with the first part of n + 1 th the term and leaving the first part of the first term. Repeated application of this process yields a new series which approaches 0 and the series of the left out first parts (‘orphans’) that converges faster than the original series.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (02) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

Possible and Impossible Series

2015 ◽  
Vol 108 (7) ◽  
pp. 560
Author(s):  
Mark MacLean
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Its Sequence

A lesson helps students discern possible relationships between an infinite series, its sequence of terms, and the sequence of partial sums.

Local contractivity of the process of a player rating variation in the Elo model with one adversary

2015 ◽  
Vol 25 (5) ◽  
Author(s):  
Vadim A. Avdeev
Keyword(s):  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Infinite Series ◽  
Limit Distribution ◽  
Rating Model ◽  
Upper Estimate

AbstractWe study the process of variation of a player rating in an infinite series of games with the same adversary in the Elo rating model. This process is shown to have a stationary distribution, an upper estimate of the rate of convergence to which is established. In a previous paper by the author, the existence of a limit distribution was proved under more stringent assumptions on the parameters of a rating model.

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1970 ◽  
Vol 22 (2) ◽  
pp. 202-208 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Cesàro Means ◽  
Riesz Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Image Position ◽  
Generalized Cesàro Means

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

On a Tauberian theorem for absolute harmonic summability

1969 ◽  
Vol 65 (1) ◽  
pp. 93-99 ◽  
Author(s):  
O. P. Varshney ◽  
Govind Prasad
Keyword(s):  
Infinite Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Image Position ◽  
Harmonic Summability

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined bywhere is given by

An approximation of stopped sums with applications in queueing theory

1987 ◽  
Vol 19 (03) ◽  
pp. 674-690 ◽  
Author(s):  
Miklós Csörgő ◽  
Paul Deheuvels ◽  
Lajos Horváth
Keyword(s):  
Rate Of Convergence ◽  
Renewal Process ◽  
Queueing Theory ◽  
Limit Theorems ◽  
Partial Sums ◽  
Probability Inequalities ◽  
Strong Approximations ◽  
Stopped Sums

We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate of convergence of some limit theorems in queueing theory.

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