EXAMPLE 5Express 16/(1 - 4x)²as a power series by differentiating the equation below. What is the radius of convergence?44(1 + 4x + 16x² + 64x³ + ...) = 4 £ (4x)"n=0(1 - 4x)SOLUTIONDifferentiating each side of the equation, we get16-= 4(4 ++ 192x2 + ...)(1 - 4x)2= 4 En=1If we wish, we can replace n by n + 1 and write the answer as16-= 4 E(1 - 4x)²n=0According to Theorem 2, the radius of convergence of the differentiated series is the same as the radius on convergence of the original series, namely, R =

solution:-step1:4(1-4x)=4∑n=0∞4xnon differentiating each side of the equation;we…

EXAMPLE 5 Express 16/(1 - 4x)2 as a power series by differentiating the equation below. What is the radius of convergence? 4 00 = 4(1 + 4x + 16x2 + 64x3 + ...) = 4 E (4x)" n=0 (1 - 4x) SOLUTION Differentiating each side of the equation, we get 16 = 4(4 + + 192x2 + ...) (1 - 4x)² = 4 E n=1 If we wish, we can replace n by n + 1 and write the answer as 16 00 = 4 E (1 - 4x)2 n=0 According to Theorem 2, the radius of convergence of the differentiated series is the same as the radius on convergence of the original series, namely, R =

EXAMPLE 5 Express 16/(1 - 4x)2 as a power series by differentiating the equation below. What is the radius of convergence? 4 00 = 4(1 + 4x + 16x2 + 64x3 + ...) = 4 E (4x)" n=0 (1 - 4x) SOLUTION Differentiating each side of the equation, we get 16 = 4(4 + + 192x2 + ...) (1 - 4x)² = 4 E n=1 If we wish, we can replace n by n + 1 and write the answer as 16 00 = 4 E (1 - 4x)2 n=0 According to Theorem 2, the radius of convergence of the differentiated series is the same as the radius on convergence of the original series, namely, R =

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 82E

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