The sum sum limits_{n=1}^{infty} frac{2 n^2+3 | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$\sum \limits_{ n =1}^{\infty} \frac{2 n ^2+3 n +4}{(2 n ) !}$

$\frac{1}{2} \sum \limits_{ n =1}^{\infty} \frac{2 n (2 n -1)+8 n +8}{(2 n ) !}$

Vedclass · Accepted Answer

$\frac{13 e }{4}+\frac{5}{4 e }-4$

Vedclass · Answer

$\sum \limits_{ n =1}^{\infty} \frac{2 n ^2+3 n +4}{(2 n ) !}$

$\frac{1}{2} \sum \limits_{ n =1}^{\infty} \frac{2 n (2 n -1)+8 n +8}{(2 n ) !}$

$\frac{1}{2} \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -2) !}+2 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -1) !}+4 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n ) !}$

$e =1+1+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \ldots$

$e ^{-1}=1-1+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}+\ldots \ldots$

$\left( e +\frac{1}{ e }ight)=2\left(1+\frac{1}{2 !}+\frac{1}{4 !}+\ldots \ldots .ight)$

$e -\frac{1}{ e }=\left(1+\frac{1}{3 !}+\frac{1}{5 !}+\ldots . .ight)$

$	ext { Now }$

$\frac{1}{2}\left(\sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -2) !}ight)+2 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n -1) !}+4 \sum \limits_{ n =1}^{\infty} \frac{1}{(2 n ) !}$

$=\frac{1}{2}\left[\frac{ e +\frac{1}{ e }}{2}ight]+2\left[\frac{ e -\frac{1}{ e }}{2}ight]+4\left(\frac{ e +\frac{1}{ e }-2}{2}ight)$

$=\frac{\left( e +\frac{1}{ e }ight)}{4}+ e -\frac{1}{ e }+2 e +\frac{2}{ e }-4$

$=\frac{13}{4} e +\frac{5}{4 e }-4$

The sum $\sum \limits_{n=1}^{\infty} \frac{2 n^2+3 n+4}{(2 n) !}$ is equal to :

Similar Questions

Solution set of equation

$\left| {1 - {{\log }_{\frac{1}{6}}}x} \right| + \left| {{{\log }_2}x} \right| + 2 = \left| {3 - {{\log }_{\frac{1}{6}}}x + {{\log }_{\frac{1}{2}}}x} \right|$ is $\left[ {\frac{a}{b},a} \right],a,b, \in N,$ then the value of $(a + b)$ is

Let $n$ be the smallest positive integer such that $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n} \geq 4$. Which one of the following statements is true?

The value of ${\log _2}.{\log _3}....{\log _{100}}{100^{{{99}^{{{98}^{{.^{{.^{{{.2}^1}}}}}}}}}}}$ is

The number of solution pairs $(x, y)$ of the simultaneous equations $\log _{1 / 3}(x+y)+\log _3(x-y)=2$ $2^{y^2}=512^{x+1}$ is

If ${\log _{\tan {{30}^ \circ }}}\left( {\frac{{2{{\left| z \right|}^2} + 2\left| z \right| - 3}}{{\left| z \right| + 1}}} \right)\, < \, - 2$ then