The sum of the series \[\frac{{{1}^{2}}}{1\cdot 2\,!}+\frac{{{1}^{2}}+{{2}^{2}}}{2\cdot 3\,!}+\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}{3\cdot 4\,!}+..+\frac{{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}}{n\cdot (n+1)\,!}+...\infty \]equals [AMU 2002]

\[\frac{1}{2}{{(e+{{e}^{-1}})}^{2}}\]

The sum of the series \[\frac{{{1}^{2}}}{1\cdot 2\,!}+\frac{{{1}^{2}}+

1.	The sum of the series \[\frac{{{1}^{2}}}{1\cdot 2\,!}+\frac{{{1}^{2}}+{{2}^{2}}}{2\cdot 3\,!}+\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}{3\cdot 4\,!}+..+\frac{{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}}{n\cdot (n+1)\,!}+...\infty \]equals [AMU 2002]
A.	\[{{e}^{2}}\]
B.	\[\frac{1}{2}{{(e+{{e}^{-1}})}^{2}}\]
C.	\[\frac{3e-1}{6}\]
D.	\[\frac{4e+1}{6}\]
Answer» D. \[\frac{4e+1}{6}\]

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The sum of the series \[\frac{{{1}^{2}}}{1\cdot 2\,!}+\frac{{{1}^{2}}+{{2}^{2}}}{2\cdot 3\,!}+\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}{3\cdot 4\,!}+..+\frac{{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}}{n\cdot (n+1)\,!}+...\infty \]equals [AMU 2002]

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