An Iterated Sum Using Calculus And/Or Algebra

Calculus Level 4

If f(n)=k=21knk!f(n) = \displaystyle \sum_{k=2}^{\infty} \frac{1}{k^nk!}, then evaluate n=2f(n)\displaystyle \sum_{n=2}^{\infty} f(n).

This problem appeared in HMMT.
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