Section10.6-Power Series

N 1 ln 1 x n 1 n0 so r1 integrationexamplecontinued

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Unformatted text preview: ln 1 0 A n 0 n 1 0 A0 A0 x n 1 ln 1 x n 1 n0 So, R = 1 Integration Example continued What about endpoints? At x = ‐ 1 : n 0 1 n 1 n 1 Series is convergent. At x = 1 : n 0 1 n 1 Series is divergent. Interval of convergence for series is [ ‐ 1 , 1 ) but series is only guaranteed to be equal to the function within R, (‐ 1 , 1). It can be shown that the series converges to the function for x in [‐1,1). n 1 x ln 1 x n 1 n 0 n on [ ‐1, 1 ) i 1 x ln 1 x Also, for x close to 0. i 1 i 0 Example Find a series representation for ln(2/3). x n 1 ln 1 x n 1 n 0 R = 1 n 1 1 1 3 ln 1 3 n 0 n 1 2 1 ln n 3 n 1 3 n Example What is the sum of the convergent series: Solution: x n 1 ln 1 x n0 n...
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