<s>[INST] Sum of series of fractions: $\frac{4}{1!}+ \frac{8}{2!}+ \frac{14}{3!} + \frac{22}{4!}+\cdots$ 
Find the sum of the series:
  $$\frac{4}{1!}+ \frac{8}{2!}+ \frac{14}{3!} + \frac{22}{4!}+\cdots$$

I am not able to understand how to proceed. The numerator terms have nothing in common which might result in an AP or GP.
Please guide me with the approach.
 [/INST] $a_n = \frac{n^2 + n + 2}{n!}\\
\sum a_n = \sum\frac{n^2 + n + 2}{n!}\\
\sum_\limits{n=1}^\infty\frac{n^2 - n}{n!} + \sum_\limits{n=1}^\infty\frac{2n}{n!} + \sum_\limits{n=1}^\infty \frac {2}{n!}\\$
$\sum_\limits{n=1}^\infty\frac{n^2 - n}{n!} = 0 + \sum_\limits{n=2}^\infty\frac{1}{(n-2)!} = \sum_\limits{n=0}^\infty\frac{1}{n!} = e\\
\sum_\limits{n=1}^\infty\frac{2n}{n!} = 2\sum_\limits{n=1}^\infty\frac{1}{(n-1)!} = 2e\\
\sum_\limits{n=1}^\infty \frac {2}{n!} = 2e- 2$
$5e-2$
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