Power series

[hgaon001]

I have that f(x) = the sum from n=0 to infinity of [((-1)^n)((x-2)^n)/(5^n)]

- im asked to fine the power series for the integral of f(x) and the interval.
-power series for f '(x)

i already have that the interval is -3<x<7 by the ratio test and testing the points. however, i dont know what to do after that.
i kno its conditionally convergent for the integral and it has the same interval for the derivative.. but i still dont know how to show it.

 

[DrMike]

The series for the integral and derivative will have the same radius of convergence, so the intervals of convergence will be either (-3,7), [-3,7), (-3,7] or [-3,7].

Once you write out the power series for the integral and derivative, plug in x=-3 and x=7 and see if the series you get converge. You'll find, after simplifying, that it's not hard to see.