My exam is coming up, I have 2 questions on infinite series. Any help is appreciated! Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1 (t^2) dt = Sigma (k=0, infinity) [(-1)^k * x^(4k+3)] / [(2k+1) (4k+3)] 0 But the part b, they also ask for the radius of convergence, how can I find it? Should I apply the ratio test to this series expansion (colored in red) to find the radius of convergence? Is there any faster way? Question 2) Suppose f(x)= x cos(x^2), find f^(4101) (0). [f^(n) (0) is the "n"th derivative evaluated at 0] Should I use the fact that "On its interval of convergence, a power series is the Taylor series of its sum" to do this question? So is it true that the power series of x cos(x^2) is EQUAL to the Taylor series of x cos(x^2) for ALL real numbers x? Is there any difference between power series and Taylor series? Thanks for your help!