Harmony in the world!

Calculus Level 5

\[\large \displaystyle \sum_{k=1}^{\infty} \dfrac{(-1)^{(k+1)}H_{k}}{k}\]

If the above sum can be expressed in the form \[ \large \dfrac{\zeta(P)}{R} - \dfrac{\ln^{2} R}{P} \; ,\] where \(P\) and \(R\) are positive integers, find \(P+R\).

Notation: \( H_n\) denotes the \(n^\text{th} \) harmonic number, \( H_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n\).

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