Came across the fact that \[ \frac{1}{96} = \sum_{k=1}^{\infty} \frac{k^2}{(k+1)(k+2)(k+3)(k+4)(k+5)} \], without any references. How do I go about proving that? I hope the answer doesn't involve the Riemann ζ function.

@christianp This looks like a good candidate for induction/limit. I would move to partial sums, then pull the sum inside the product for what is here the denominator, and work out a formula in terms of the number of terms n in the partial sum, then take the limit of that formula as n increases.

@christianp *frowns* ok, the sum doesn't pull inside /directly/, but you get what i mean; it's only a five term expansion at worst. Definitely go from (inductively proved) finite sum formulas on the individual sub-functions of k.