TITLE: $\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}(\zeta(2)-H_{k,2})-H_n\right)=1$
QUESTION [5 upvotes]: I found this limit in a book, without any explanation:
$$\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}(\zeta(2)-H_{k,2})-H_n\right)=1$$
where $H_{k,2}:=\sum_{j=1}^k\frac1{j^2}$. However Im unable to find the value of this limit from myself. After some work I get the equivalent expression
$$\lim_{n\to\infty}\sum_{k=0}^{n-1}\sum_{j=k}^\infty\frac1{(j+1)^2(j+2)}$$
but anyway Im stuck here. Can someone show me a way to compute this limit? Thank you.
UPDATE: Wolfram Mathematica computed it value perfectly, so I guess there is some integral or algebraic identity from where to calculate it.

REPLY [2 votes]: Considering your last expression $$a_n=\sum_{k=0}^{n-1}\sum_{j=k}^\infty\frac1{(j+1)^2(j+2)}$$
$$\sum_{j=k}^\infty\frac1{(j+1)^2(j+2)}=\psi ^{(1)}(k+1)-\frac{1}{k+1}$$ making
$$a_n=n \,\psi ^{(1)}(n+1)$$ the expansion of which being
$$a_n=1-\frac{1}{2 n}+\frac{1}{6 n^2}+O\left(\frac{1}{n^4}\right)$$