Outline a method for calculating S_k = Σ n^k/n! for n=1 to infinity and compute the first few terms of S_k for natural numbers k.

(In reply to Solution by Praneeth)

You have, for example

S(2) = C(1,0) S(0)+ C(1,1) S(1) = e-1+e=2e-1

However, the first four terms of S(2) are:

1^2/1! + 2^2 / 2! + 3^2 / 3! + 4^2 / 4!

= 1 + 2 + 3/2 + 16/24

= 1 + 2 + 3/2 + 2/3

which totals 5 + 1/6, which already exceeds  2e - 1. None of the terms are negative, so the infinite sum must be larger, and I have computed as 2e. Likewise the remaining ones are all integral multiples of e without any integer subtracted.

Posted by Charlie on 2008-02-20 11:04:57