For the following indefinite integral, find the full power series centered at x=0 and give the first 5 nonzero terms of the power series and the open interval of convergence. f(t)=∫(arctan(2x))/x dx

Answer 1

Isn't it Gregory's formula that holds in the closed interval -1≤2x≤1

tan^-12x=(2x)-(2x)³/3+(2x)⁵/5-(2x)⁷/7+(2x)⁹/9-/+...

dividing by x is not a big deal

then integrating is not a big deal

I suppose ratio test will be used

for interval of convergence