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

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


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The solution

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  oo        
____        
\   `       
 \      2*n 
  \    x    
  /   ------
 /    (2*n)!
/___,       
n = 1
∑n=1∞x2n(2n)!\sum_{n=1}^{\infty} \frac{x^{2 n}}{\left(2 n\right)!} 
Sum(x^(2*n)/factorial(2*n), (n, 1, oo))
The radius of convergence of the power series
Given number:
x2n(2n)!\frac{x^{2 n}}{\left(2 n\right)!}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=1(2n)!a_{n} = \frac{1}{\left(2 n\right)!}
and
x0=0x_{0} = 0
,
d=2d = 2
,
c=1c = 1
then
R2=lim⁡n→∞∣(2n+2)!(2n)!∣R^{2} = \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|
Let's take the limit
we find
R2=∞R^{2} = \infty
R=∞R = \infty
The answer [src] 
 2 /  2    2*cosh(x)\
x *|- -- + ---------|
   |   2        2   |
   \  x        x    /
---------------------
          2
x2(2cosh⁡(x)x2−2x2)2\frac{x^{2} \left(\frac{2 \cosh{\left(x \right)}}{x^{2}} - \frac{2}{x^{2}}\right)}{2} 
x^2*(-2/x^2 + 2*cosh(x)/x^2)/2

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