Power series to approximate definite integral

So i need to approximate the integral from 0 to 0.2 of
with error less than 1/1000.

so i let f(x) = x/(1-x)

= summation n=0 to infinity ( x^(n+1))

so i take the integral of the summation to get:

summation n=0 to infinity ( x^(n+2)/(n+2) )

If the series was alternating I would know how to calculate it with error to 1/1000 but I don't know how to do this if it isn't alternating.

There are various forms for the remainder term, such as LaGrange
Taylor's theorem - Wikipedia, the free encyclopedia

And you can check your work easily since



and I get as the answer/limit.