# Power series to approximate definite integral

• Mar 24th 2011, 03:12 PM
JimmyRP
Power series to approximate definite integral
So i need to approximate the integral from 0 to 0.2 of $x/(1-x)$
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.
• Mar 24th 2011, 09:18 PM
matheagle
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

${x\over 1-x}=-1+{1\over 1-x}$

and I get $-.2-\ln (.8)\approx .023143551$ as the answer/limit.