Limit of $\int_0^1 \frac{x^n-x^{2n}}{1-x}\text{d}x$ I have to find the value of  $$\lim_{n\to\infty} \int_0^1 \frac{x^n-x^{2n}}{1-x}\text{d}x$$ with $n\in\mathbb{N}$. The result is $\ln 2$ but i find no way to prove it. I thought that $$H_n=\int_0^1 \frac{1-x^n}{1-x}\text{d}x$$ would have helped but I came up with nothing. The fact that the result is the the same as the sum of alternating harmonic series made me think i should find a way to make a $(-1)^k$ pop out from somewhere but all my attempts failed… Any idea?

Your integral is $H_{2n}-H_n$ which using known behavior of $H_n$ at large $n$, that is $H_{n\gg 1} = \ln n + C$, gives $$\lim_{n\to\infty}I=\lim_{n\to\infty}(H_{2n}-H_{n})=\ln 2.$$ 
