Question

find power series for (1/(1+x^2)). use this power series to prove that the taylor series centered at x=0 for actan(x) is

x -x^3/3 +x^5/5 -... (-1)^n ((x^2n+1)/(2n+1))...

Answer 1

fix) = x Power series for f(x) using the formula 1 x no Now taylor series for f(x) = arctonn. flui= tantu 26x) = 1 at u=o f'(o) = 1 1"(X)= (-(1+x2)*(3x) = 1+x2 (1+x2) f" (0) = 0 f (x) = (-1) (+2) (1+x) (2) + (-1) (-2) (1 + 4%) 3 (2x) f (x) = – 2 (1+x²)2 + 2(2x3² (1+x) - 3 +") -2. "*)= (-2)(-a) ()+x+)?c9x) + 2 (3)(2x)(?) (1+x°) 3 + 3(2x) * (-3) (1+x) * (24) Now taylor series atx=0 70x) = 4(0)+ xfco) + x?%+***"0+ --- +14) = 0+x61+ (9) + - x-x+*+- (-" 2n +