find the power series and the interval of convergence

Question

use that fact that 1/1-x= sumx^n with interval of convergence (-1,1)

Answer 1

For the first one, notice that 1/(1-x)² is the derivative of 1/(1-x).

So, differentiate the power series for 1/(1-x) term by term and then multiply each term by x⁵.

For the other one, The derivative of Tan^-1x is 1 / (1+x²) = 1 / (1 - (-x²))

So, start with the power series for 1/(1-x). Then replace x by -x². Next, integrate term by term to get the power series for Tan^-1x. Finally replace x by x².