b
\(\sum \limits_{ n =1}^{\infty} \frac{2 n ^2+3 n +4}{(2 n ) !}\)

\(\frac{1}{2} \sum \limits_{ n =1}^{\infty} \frac{2 n (2 n -1)+8 n +8}{(2 n ) !}\)

\(\frac{1}{2} \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -2) !}+2 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -1) !}+4 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n ) !}\)

\(e =1+1+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \ldots\)

\(e ^{-1}=1-1+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}+\ldots \ldots\)

\(\left( e +\frac{1}{ e }\right)=2\left(1+\frac{1}{2 !}+\frac{1}{4 !}+\ldots \ldots .\right)\)

\(e -\frac{1}{ e }=\left(1+\frac{1}{3 !}+\frac{1}{5 !}+\ldots . .\right)\)

\(\text { Now }\)

\(\frac{1}{2}\left(\sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -2) !}\right)+2 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -1) !}+4 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n ) !}\)

\(=\frac{1}{2}\left[\frac{ e +\frac{1}{ e }}{2}\right]+2\left[\frac{ e -\frac{1}{ e }}{2}\right]+4\left(\frac{ e +\frac{1}{ e }-2}{2}\right)\)

\(=\frac{\left( e +\frac{1}{ e }\right)}{4}+ e -\frac{1}{ e }+2 e +\frac{2}{ e }-4\)

\(=\frac{13}{4} e +\frac{5}{4 e }-4\)