Section10.6-Power Series

xn n ln 1 x n 1 1 n n n 1 n 1 1 n adjusttheindex

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Unformatted text preview: 1 1 n 1 for x in [ ‐ 1, 1 ) (Previously derived.) xn n ln 1 x n 1 1 n n n 1 n 1 1 n (Adjust the index.) ln 1 1 (Plug in x = ‐ 1.) n 1 ln 2 n 1 1 n Quiz Find the sum of the series n 0 A) ln 1 4 4 B) ln 1 C) ln 3 4 4 1 n 1 n 1 Quiz Find the sum of the series 5 A) ln 4 B) ln 6 C) 5 6 ln 5 1 n0 n 1 1 5n 1 n 1 Example tan 1 x Find a power series representation for First, find a power series representation for 1 1 x2 1 1 1 x2 1 x2 x n0 2n 1 n 1 x 2 n 1 x 2 n0 Then, integrate to find power series representation for tan 1 x Example continued 1 dx 2 1 x n 2n A 1 x dx A 1 x 2 x 4 x 6 dx n 0 tan 1 x A x 2 n 1 x3 x5 x 7 tan 1 x A 1 A x 2n 1 357 n 0 n Solve for A using x = 0: 03 05 0 7 tan 1 0 A 0 357 A0 Example continued x 2 n 1 tan x 1 2n 1 n 0 1 n R = 1 Note: It can be proven that this series converges to the function within the interval [‐1,1]. Thus 12 n 1 tan 1 1 1 2n 1 n 0 n 1 4 n 0 2n 1 n 444 111 4 1 4 357 357 TryIt.5 444 4 357 How many terms need to be added to approximate π to within 0.001? Quiz Find the sum of the series A) No sum (divergent) B) C) 6 3 2 n 1 3 n 1 2 n 1 3 2n 1 n0...
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