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Idea: use of Taylor series in regions of x-space where the series if divergent. One can devise a rational polynomial expression that is valid for a much wider range of x than the original Taylor series – yet the rational approximation (known as a Pade approximant) uses ONLY the Taylor series coefficients that lead to a divergent Taylor series!! (A) Consider the function f(x) that is defined by In(1+x) S(x)= Show that the Taylor series expansion of f(x) about x = 0 is (-1)"x" n+1 n=0

Idea: use of Taylor series in regions of x-space where the series if divergent. One can devise a rational polynomial expression that is valid for a much wider range of x than the original Taylor series – yet the rational approximation (known as a Pade approximant) uses ONLY the Taylor series coefficients that lead to a divergent Taylor series!! (A) Consider the function f(x) that is defined by In(1+x) S(x)= Show that the Taylor series expansion of f(x) about x = 0 is (-1)"x" n+1 n=0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 49E
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Idea: use of Taylor series in regions of x-space where the series if divergent. One can devise a rational polynomial expression that is valid for a much wider range of x than the original Taylor series – yet the rational approximation (known as a Pade approximant) uses ONLY the Taylor series coefficients that lead to a divergent Taylor series!! (A) Consider the function f(x) that is defined by In(1+x) S(x)= Show that the Taylor series expansion of f(x) about x = 0 is (-1)"x" n+1 n=0
Transcribed Image Text:Idea: use of Taylor series in regions of x-space where the series if divergent. One can devise a rational polynomial expression that is valid for a much wider range of x than the original Taylor series – yet the rational approximation (known as a Pade approximant) uses ONLY the Taylor series coefficients that lead to a divergent Taylor series!! (A) Consider the function f(x) that is defined by In(1+x) S(x)= Show that the Taylor series expansion of f(x) about x = 0 is (-1)"x" n+1 n=0
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