Change in change in harmony

Calculus Level 4

\[ \large \sum_{n=1}^\infty \left [ H_n^{(2)} \left ( \dfrac1{n^2} - \dfrac1{(n+1)^2} \right) \right ] \]

If the value of the series above is in the form of \( \dfrac{\pi^ A}B \), where \(A\) and \(B\) are integers, find \(A+B\).

Notation: \( H_n^{(m)} \) denotes the generalized harmonic number, \(\displaystyle H_n^{(m)} = \sum_{k=1}^n \dfrac1{k^m} \).

Inspirations: First link, Second link.

There is a much simpler way to solve this than the two other questions.

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