I'm studying for a qualifying exam and I'm stuck on this problem from Bass's "Real Analysis for Graduate Students" (Exercise 7.14). It asks us to prove that
$$\sum_{k=1}^\infty\frac{1}{(p+k)^2}=-\int_0^1\frac{x^p}{1-x}\log(x)dx$$ for $p>0$.
Note that this exercise comes from the chapter that introduces the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem. My issue is probably that I don't immediately see how to apply any of these theorems to this particular problem. I have tried playing around with Feynman's trick and log series, but haven't made any notable progress. I've been stuck here for awhile so any help is appreciated.
Best Answer
First, expand $1/(1-x)$ into its geometric series to get $$-\int_0^1\sum_{k\ge 0} x^mx^p\log x\,dx$$ Now, consider the partial sums $$S_N=\sum_{k=0}^Nx^k=\frac{1-x^{N+1}}{1-x}$$ We can easily show that $S_N\le S_{N+1}$, as $S_{N+1}-S_N=x^{N+1}\ge 0$ as $x\in[0,1]$. By the monotone convergence theorem, \begin{align} -\int_0^1\sum_{k\ge 0} x^k x^p\log x\,dx&=-\int_0^1\lim_{N\to\infty}\sum_{k= 0}^N x^k x^p\log x\,dx \\ &=-\lim_{N\to\infty}\int_0^1\sum_{k= 0}^N x^k x^p\log x\,dx \tag{1} \\ &=-\lim_{N\to\infty}\sum_{k= 0}^N\int_0^1 x^k x^p\log x\,dx \\ &=-\sum_{k\ge 0}\int_0^1 x^k x^p\log x\,dx \\ &=\sum_{k\ge 0}\frac{1}{(k+p+1)^2} \tag{2}\\ &=\sum_{k\ge 1}\frac{1}{(k+p)^2} \\ \end{align} Where the monotone convergence theorem was used in $(1)$ and integration by parts in $(2)$.