Question

1 Problem 7 We know that we can expand as a power series for -1 < < 1. 1+2 Follow the given steps to manipulate this power series to derive the power series representation for f(x) = tan-(2) centered at a = 0. • Make the appropriate substitution to find a power series for g(x) 1/(1 + x2). • Either integrate or differentiate the previous power series to find a power series for f(x) = tan-'(x). Has the radius of convergence changed? Has the interval of convergence changed? After you have derived the power series expansion for f, write the answer in series notation in the box provided. (Show all details.)

Answer 1

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to hare , n! (-) anol O NORD let R be the sadius o Co mwengence of the poweho genios, به اسم lem antl an slim no (-1)^+1+1 2 (n+1)-1 (-1) Qual mt lim 2n-1 anta lim an-1 antl lim 12 - an REI so, the genoeg corn for inle Now , R=1 the fon sessted becomes Ź (1) **! mal an+ 1 So, it convenges by Libreites nop I { (-1) २१-1 feat

foro x=-1 the denies become a nal Ź (-1) **' (649" m21 22-1 DO amti Z (-1) an an-l mzi N 1 2 an-l nal diverges So interval of convergence is (-7,1] and madius of convergence remain unchanged. 9