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Free Response 11 The function is defined by the power series f(x) = 1 + (x+1)+ (x + 1)² ++ (x + 1)² + ... = -=[(x+1)" n=0 (x+1)*. -/

Free Response 11 The function is defined by the power series f(x) = 1 + (x+1)+ (x + 1)² ++ (x + 1)² + ... = -=[(x+1)" n=0 (x+1)*. -/

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 50E
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Free Response 11 The function is defined by the power series f(x) = 1 + (x+1)+ (x + 1)² ++ (x + 1)² + ... = -=[(x+1)" n=0 (x+1)*. -/<x+1 <1 (22x20 for all real numbers x for which the series converges. (a) Find the interval of convergence of the power series for f. Justify your answer. (b) The power series above is the Taylor series for f about x = -1. Find the sum of the series for f. (c) Let g be the function defined by g(x) = f(t) dt. Find the value of g(-2), if it exists, or explain why g(-1) cannot be determined. (d) Leth be the function defined by h(x) = f(x²-1). Find the first three nonzero terms and the general term of the Taylor series for h about x = 0, and find the value of h 8 h (2).
Transcribed Image Text:Free Response 11 The function is defined by the power series f(x) = 1 + (x+1)+ (x + 1)² ++ (x + 1)² + ... = -=[(x+1)" n=0 (x+1)*. -/<x+1 <1 (22x20 for all real numbers x for which the series converges. (a) Find the interval of convergence of the power series for f. Justify your answer. (b) The power series above is the Taylor series for f about x = -1. Find the sum of the series for f. (c) Let g be the function defined by g(x) = f(t) dt. Find the value of g(-2), if it exists, or explain why g(-1) cannot be determined. (d) Leth be the function defined by h(x) = f(x²-1). Find the first three nonzero terms and the general term of the Taylor series for h about x = 0, and find the value of h 8 h (2).
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