Harmony

Calculus Level 5

If

$\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^2}$

can be expressed in the form $$\dfrac{a\pi^k}{b}$$, where $$a, b$$, and $$k$$ are positive integers with $$a$$ and $$b$$ coprime, find $$a+b+k$$.

Notation: $$\displaystyle H_n^{(s)} = \sum_{m=1}^n \frac{1}{m^s}$$.

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