Sum of series 7.

\(\displaystyle\;\; \sum_{n=0}^{\infty} \frac{5n+1}{(2n+1)!}\)

\(\displaystyle \text{Above expression can be written as }\)

\(\displaystyle \sum_{n=0}^{\infty} \frac{5n+1}{(2n+1)!} = \sum_{n=0}^{\infty}\left(\frac{(2n+1)}{(2n+1)!}+\frac{3n}{(2n+1)!}\right)\)

\(\displaystyle \text{This can also be written as }\)

\(\displaystyle \Rightarrow \sum_{n=0}^{\infty}\frac{(2n+1)}{(2n+1)!}+\sum_{n=0}^{\infty}\frac{3n}{(2n+1)!}\)

\(\displaystyle \text{Again further simplifying it we will get }\)

\(\displaystyle \Rightarrow \sum_{n=0}^{\infty}\frac{1}{(2n)!}+3\sum_{n=0}^{\infty}\frac{n}{(2n+1)!}\cdots \cdots\;\;\;\;(1)\)

\(\displaystyle \text{We have a series } \;\;\;\; \cosh x = 1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}\cdots\)

\(\displaystyle \text{We will use above series in (1) we will get }\)

\(\displaystyle \Rightarrow \cosh 1 +3\sum_{n=0}^{\infty}\frac{n}{(2n+1)!}\cdots \cdots\;\;\;\;(2)\)

\(\displaystyle \text{Now further simplifying (2) we will get }\)

\(\displaystyle \Rightarrow \cosh 1 +3\sum_{n=0}^{\infty}\left(\frac{1}{(2n)!}-\frac{1}{(2n+1)!}\right)\)

\(\displaystyle \Rightarrow \cosh 1 +3\sum_{n=0}^{\infty}\frac{1}{(2n)!}-3\sum_{n=0}^{\infty}\frac{1}{(2n+1)!}\)

\(\displaystyle \Rightarrow \cosh 1 +3\cosh 1-3\sum_{n=0}^{\infty}\frac{1}{(2n+1)!}\cdots \cdots\;\;\;\;(3)\)

\(\displaystyle \text{Again we have one more series expansion}\)

\(\displaystyle \sinh x =x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}\cdots )\)

\(\text{Using this in (3) we get }\)

\(\displaystyle \Rightarrow \cosh 1 +3\cosh 1-3\sinh 1 \cdots \cdots (4)\)

\(\text{Hyperbolic functions in exponential form are}\)

\(\displaystyle \sinh 1 =\frac{e-e^{-1}}{2}\;\;\;\text{and}\;\;\; \cosh 1 =\frac{e+e^{-1}}{2}\)

\(\text{Using above exponential forms in (4) we get }\)

\(\displaystyle \Rightarrow \frac{e+e^{-1}}{2} +3\left(\frac{e+e^{-1}}{2}\right) 1-3\left(\frac{e+e^{-1}}{2}\right)\)

\(\displaystyle \Rightarrow \frac{e}{2} +2e^{-1}\)
\(\displaystyle \text{ANSWER:}\;\;\;\; \frac{e}{2} +\frac{2}{e}\)