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The evaluation of this integral is a quaint smorgasbord of calculus techniques. All that's missing is some nasty variable substitution.

The rightfully skeptical reader may be assured that the grouping of terms in the sum evaluation is valid. For the nth partial sum is equal to the nth partial sum of the geometric series minus n/exp(n+1), and thus we can write the sequence of partial sums as the difference of two sequences: the partial sums of the geometric sequence which converges to 1/(e-1), and the sequence {n/exp(n+1)} which converges to 0. Analysis'd!