Infinite series and power series

2021 ◽  
pp. 292-332
Keyword(s):  
Power Series ◽  
Infinite Series

Bernstein Power Series

1964 ◽  
Vol 16 ◽  
pp. 241-252 ◽  
Author(s):  
E. W. Cheney ◽  
A. Sharma
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Approximation Theorem ◽  
Bernstein Polynomials ◽  
Starting Point ◽  
Image Position ◽  
Weierstrass Approximation Theorem

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomialsare constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.

scholarly journals A Natural Series for the Natural Logarithm

10.37236/880 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Oliver T. Dasbach
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Mahler Measure ◽  
Series Expansions ◽  
Natural Logarithm ◽  
Similar Series ◽  
Power Series Expansions ◽  
Gromov Norm ◽  
Logarithm Function ◽  
Natural Series

Rodriguez Villegas expressed the Mahler measure of a polynomial in terms of an infinite series. Lück's combinatorial $L^2$-torsion leads to similar series expressions for the Gromov norm of a knot complement. In this note we show that those formulae yield interesting power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of $4$.

XXVII.—Expansions of Lamé Functions into Series of Legendre Functions

1948 ◽  
Vol 62 (3) ◽  
pp. 247-267
Author(s):  
A. Erdélyi
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Infinite Series ◽  
Recurrence Relations ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Jacobi Series ◽  
Characteristic Numbers ◽  
Transcendental Equations

Summary28. This paper contains the investigation of certain properties of periodic solutions of Lamé's differential equation by means of representation of these solutions by (in general infinite) series of associated Legendre functions. Terminating series of associated Legendre functions representing Lamé polynomials have been used by E. Heine and G. H. Darwin. The latter used them also for numerical computation of Lamé polynomials. It appears that infinite series of Legendre functions representing transcendental Lamé functions have not been discussed previously. In two respects these series seem to be superior to the generally used power-series and Fourier-Jacobi series, (i) They are convergent in some parts of the complex plane of the variable where both power-series and Fourier-Jacobi series diverge, (ii) They permit by simply replacing Legendre functions of first kind by those of second kind, to deal with Lamé functions of second kind as well as Lamé functions of first kind (§ 15).In §§ 2 and 8 of the present paper the series are heuristically deduced from the integral equations satisfied by periodic Lamé functions. Inserting the series found heuristically, with unknown coefficients, into Lamé's differential equation, recurrence relations for the coefficients are obtained (§§ 9–12). These recurrence relations yield the (in general transcendental) equations in form of (in general infinite) continued fractions for the determination of the characteristic numbers. The convergence of the series can be discussed completely.There are altogether forty-eight different series. Every one of the eight types of Lamé polynomials is represented by six different series (see table in § 13). There are interesting relations (§ 14) between series representing the same function.Next infinite series representing transcendental Lamé functions are discussed. It is seen that transcendental Lamé functions are only simply-periodic (§§ 18, 19). Lamé functions of real (§§ 20–22) and imaginary (§§ 23-24) period are represented by series of Legendre functions the variables of which are different in both cases.The paper concludes with a brief discussion of the most important limiting cases, and a short mention of other types of series of Legendre functions representing Lamé functions.

A Tauberian theorem for absolute summability

1970 ◽  
Vol 67 (2) ◽  
pp. 321-326 ◽  
Author(s):  
G. Das
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Tauberian Theorem ◽  
Power Series Expansion ◽  
Binomial Coefficient ◽  
Absolute Summability ◽  
Partial Sums ◽  
Function Of Bounded Variation ◽  
Image Position ◽  
The Given

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by

scholarly journals On 'Best' Rational Approximations to $\pi$ and $\pi+e$

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Continued Fraction ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

scholarly journals Fibonacci Series from Power Series

2020 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Gamma Function ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Euler Numbers ◽  
Lucas Numbers ◽  
Fibonacci Series ◽  
Digamma Function ◽  
Inverse Trigonometric Functions

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

scholarly journals On Rational Approximations to $\pi+e$: 920/157 Is the 'Best' Rational Approximation to $\pi+e$ with a Denominator of at Most Three Digits

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations ◽  
Best Rational Approximation

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. This is not so surprising when one considers the empirical computations around these two rational approximations to $\pi$. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, $920/157$ turns out to be the only rational number of this type.

scholarly journals Some interesting series arising from the power series expansion of(sin-1x)q

2005 ◽  
Vol 2005 (14) ◽  
pp. 2329-2336 ◽  
Author(s):  
Habib Muzaffar
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Infinite Series ◽  
Power Series Expansion ◽  
Series Expansions ◽  
Power Series Expansions

Starting from the power series expansions of(sin-1x)q, for1≤p≤4, formulae are obtained for the sum of several infinite series. Some of these evaluations involveζ(3).

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