Why is my method incorrect?
$$\lim_{n\to\infty} I_n=\int_{-1} ^{+1} |x|(1+\sum_{r=1} ^{2n} \frac{x^r}{r})$$
Since $1>|x|$ and $\lim_{n\to\infty}$ I used the sum for an infinite geometric progression to obtain the sum as $$ I_n=\int_{-1} ^{+1} |x|(\frac{1}{1-x})$$
Which is equal to
$$ I_n=2\int_{0} ^{+1} \frac{x}{1-x}$$
$$ I_n=2\int_{0} ^{+1} \frac{1-x}{x}$$
$$I_n=2[ \ln x -x]_{0} ^{1}$$
This is tending to infinity so my method is definitely wrong but why?
It isn't quite a geometric series, it is in fact the series who's term is the integral of a geometric series.
I.e. Let $f(x) = \sum_{r=1}^{2n}\frac{x^r}{r}$.
$f'(x) = \sum_{r=1}^{2n}x^{r-1} = \frac{(1-x^{2n})}{1-x}$. You can sum this using geometric series and then integrate it to find $f(x)$.