$=\frac{(2 n+1)^{2}+20 n+39}{4 \cdot(2 n+1) !}$
$=\frac{(2 n+1)^{2}+(2 n+1) \cdot 10+29}{4(2 n+1) !}$
$=\frac{1}{4}\left[\frac{(2 n+1)^{2}}{(2 n+1)(2 n) !}+\frac{(2 n+1) 10}{(2 n+1)(2 n) !}+\frac{29}{(2 n+1) !}\right]$
$=\frac{1}{4}\left[\frac{2 n+1}{(2 n) !}+\frac{10}{(2 n) !}+\frac{29}{(2 n+1) !}\right]$
$=\frac{1}{4}\left[\frac{1}{(2 n-1) !}+\frac{11}{(2 n) !}+\frac{29}{(2 n+1) !}\right]$
$S_{1}=\frac{1}{1 !}+\frac{1}{3 !}+\frac{1}{5 !}+\ldots=\frac{e-\frac{1}{e}}{2}$
$S_{2}=11\left[\frac{1}{2 !}+\frac{1}{4 !}+\frac{1}{6 !}+\ldots\right]=11\left[\frac{e+\frac{1}{e}-2}{2}\right]$
$S_{3}=29\left[\frac{1}{3 !}+\frac{1}{5 !}+\frac{1}{7 !}+\ldots\right]=29\left[\frac{e-\frac{1}{e}-2}{2}\right]$
Now, $S =\frac{1}{4}\left[ S _{1}+ S _{2}+ S _{3}\right]$
$=\frac{1}{4}\left[\frac{ e }{2}-\frac{1}{2 e }+\frac{11 e }{2}+\frac{11}{2 e }+\frac{29 e }{2}-\frac{29}{2 e }-4\right]$
$=\frac{41 e }{8}-\frac{19}{8 e }-10$
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