Have You Heard Of Euler's Sum?

Calculus Level 5

n=1Hnn5=πAB(ζ(D))CE \large \sum_{n=1}^\infty \dfrac{H_n}{n^5} = \dfrac{\pi^A}{B} - \dfrac{(\zeta (D))^C}{E}

If the equation above holds true for positive integers A,B,C,DA,B,C,D and EE, find A+B+C+D+EA+B+C+D+E.

Notations:

  • Hn H_n denotes the nthn^\text{th} harmonic number, Hn=1+12+13++1n H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n.

  • ζ()\zeta(\cdot) denotes the Riemann zeta function.

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