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

The solution

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

$$\sum_{n=1}^{\infty} \frac{x^{2 n}}{\left(2 n\right)!}$$

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

The radius of convergence of the power series

Given number:
$$\frac{x^{2 n}}{\left(2 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(2 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 2$$
,
$$c = 1$$
then
$$R^{2} = \lim_{n \to \infty} \left|{\frac{\left(2 n + 2\right)!}{\left(2 n\right)!}}\right|$$
Let's take the limit
we find
$$R^{2} = \infty$$
$$R = \infty$$

The answer [src]

 2 /  2    2*cosh(x)\
x *|- -- + ---------|
   |   2        2   |
   \  x        x    /
---------------------
          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

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Identical expressions

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

The solution

Examples of finding the sum of a series