taylor expansion and limit of a series??

$f(x)=\int_0^xtan^{-1}tdt$

what is the taylor expansion about the origin of this function?
and how do i use this to get the limit of the series

$1-\frac{1}{2}-\frac {1}{3}+\frac {1}{4}+\frac {1}{5}-\frac{1}{6}-\frac {1}{7}.......$

i could get the limit by using concepts like rearranging the terms and got a different limit since it is a conditionally convergent series and can be made to converge to any real number.but how do i get the limit using this taylor expansion.please somebody help?
Answer to the second part is $\frac {\pi}{4}-\frac {log2}{2}$