Sum of series x^(2n)/n!

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  oo      
____      
\   `     
 \     2*n
  \   x   
  /   ----
 /     n! 
/___,     
n = 1

\sum_{n=1}^{\infty} \frac{x^{2 n}}{n!}

Sum(x^(2*n)/factorial(n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{x^{2 n}}{n!}

It is a series of species

a_{n} \left(c x - x_{0}\right)^{d n}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}

In this case

a_{n} = \frac{1}{n!}

and

x_{0} = 0

d = 2

c = 1

then

R^{2} = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|

R^{2} = \infty

R = \infty

The answer [src]

   /        / 2\\
   |        \x /|
 2 |  1    e    |
x *|- -- + -----|
   |   2      2 |
   \  x      x  /

x^{2} \left(\frac{e^{x^{2}}}{x^{2}} - \frac{1}{x^{2}}\right)

x^2*(-1/x^2 + exp(x^2)/x^2)