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Suppose that E anx" has radius of convergence R> 0. Let f(x) = E anx". Then n=0 n=0 | f(2) dr = E bn+1 an n +1 Σ an+1 an n+1 n=0 n=0 for all a, b e (-R, R). The series E ana" and have the same radius of convergence. n+1 n=0 n=0 If a power series anx" converges at x = 4, then it converges absolutely at x = 4. n=0

Suppose that E anx" has radius of convergence R> 0. Let f(x) = E anx". Then n=0 n=0 | f(2) dr = E bn+1 an n +1 Σ an+1 an n+1 n=0 n=0 for all a, b e (-R, R). The series E ana" and have the same radius of convergence. n+1 n=0 n=0 If a power series anx" converges at x = 4, then it converges absolutely at x = 4. n=0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose that E anx" has radius of convergence R > 0. Let f(x) =E anx". Then n=0 n=0 fn+1 An n + 1 an+1 An n +1 f(x) dx = > n=0 n=0 for all a, b e (-R, R). The series E anx" and an have the same radius of convergence. n=0 n+1 n=0 If a power series E anx" converges at x = 4, then it converges absolutely at r = 4. n=0
Transcribed Image Text:Suppose that E anx" has radius of convergence R > 0. Let f(x) =E anx". Then n=0 n=0 fn+1 An n + 1 an+1 An n +1 f(x) dx = > n=0 n=0 for all a, b e (-R, R). The series E anx" and an have the same radius of convergence. n=0 n+1 n=0 If a power series E anx" converges at x = 4, then it converges absolutely at r = 4. n=0
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