A Harmonic Sum

Calculus Level 5

\[ \displaystyle \sum_{n=1}^\infty \frac { (H_n )^2}{2^n} = \frac {\pi ^a}{b} + \log ^2 (c) \]

Let \(H_n\) denote the \(n^{\text{th}} \) harmonic number such the above series is satisfied for positive integers \(a,b,c\).

Find \(a+b+c\).

Details and Assumptions

  • \(H_n = 1 + \frac 1 2 + \frac 1 3 + \ldots \frac 1 n \) for \(n = 1,2,3, \ldots \)

  • \( \log \) is a natural logarithm

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