# Compute $\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{H_k}{k}-\frac{1}{2}(\ln n+\gamma)^2\right)$

Compute

$$\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{H_k}{k}-\frac{1}{2}(\ln n+\gamma)^2\right)$$ where $\gamma$ - Euler's constant.

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as always: what is the source of the problem, and what sort of material is in the book just before it? – Will Jagy Jan 15 '13 at 17:03
@WillJagy: it comes from a collection of calculus problems of all kinds that I and my colleagues have gathered. I don't know what is the source of it. – user 1618033 Jan 15 '13 at 17:13
Please, $H_k=?$ – MathOverview Jan 15 '13 at 17:23
– user 1618033 Jan 15 '13 at 17:24
Mathematica calculated it to $\pi^2 \over 12$ – Santosh Linkha Jan 15 '13 at 17:31

@Chris'ssister: The manipulation in the first part (getting to $H^2_n + H^{(2)}_n$) is a trick that works any time you're essentially adding up the upper (or lower) triangular part of a symmetric square array of numbers. The details are in Concrete Mathematics, pp. 36-37. It's a useful trick to know. – Mike Spivey Jan 15 '13 at 17:38