<s>[INST] Find the sum of the series $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...$upto n terms Find the sum of the series $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...$upto n terms.
Based on my experience I presumed it is a telescopic function but not able to convert it into telescopic function.
 [/INST] Let $S_n=\sum_{k=1}^n\frac{k+2}{k!+(k+1)!+(k+2)!}$.
$$
\begin{align*}
S_n & =\sum_{k=1}^n\frac{k+2}{k!+(k+1)!+(k+2)!} \\
& =\sum_{k=1}^n\frac{k+2}{k![1+(k+1)+(k+1)(k+2)]} \\
& =\sum_{k=1}^n\frac{k+2}{k![(k+2)+(k+1)(k+2)]} \\
& =\sum_{k=1}^n\frac{k+2}{k!(k+2)^2} \\
& =\sum_{k=1}^n\frac{k+1}{(k+2)!} \\
& =\sum_{k=1}^n\frac{(k+2)-1}{(k+2)!} \\
& =\sum_{k=1}^n\left[\frac{1}{(k+1)!}-\frac{1}{(k+2)!}\right] \\
& =\frac{1}{2!}-\frac{1}{(n+2)!} \\
\end{align*}
$$
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