<s>[INST] Evaluating the sum : $\;\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$ How to evaluate this sum?
$$\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$$
Please give some technique. Binomial not working.
 [/INST] Some details regarding convergence aside, you have
$$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$
so
$$x e^x = \sum_{k=0}^\infty \frac{x^{k+1}}{k!}$$
and
$$ \int_0^t x e^x \, dx = \sum_{k=0}^\infty \frac{1}{k+2} \frac{t^{k+2}}{k!}.$$
Thus
$$\sum_{k=1}^\infty \frac{1}{k+2} \frac{1}{k!} = \int_0^1 x e^x \, dx - 1.$$
The last integral is easy to evaluate.
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