Sum of series x^(4n)/(4n)!

The solution

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  oo        
____        
\   `       
 \      4*n 
  \    x    
  /   ------
 /    (4*n)!
/___,       
n = 0

$$\sum_{n=0}^{\infty} \frac{x^{4 n}}{\left(4 n\right)!}$$

Sum(x^(4*n)/factorial(4*n), (n, 0, oo))

The radius of convergence of the power series

Given number:
$$\frac{x^{4 n}}{\left(4 n\right)!}$$
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
$$a_{n} = \frac{1}{\left(4 n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 4$$
,
$$c = 1$$
then
$$R^{4} = \lim_{n \to \infty} \left|{\frac{\left(4 n + 4\right)!}{\left(4 n\right)!}}\right|$$
Let's take the limit
we find
$$R^{4} = \infty$$
$$R = \infty$$

The answer [src]

  ___   ____   ___ /besseli(-1/2, x)   besselj(-1/2, x)\
\/ 2 *\/ pi *\/ x *|---------------- + ----------------|
                   \       4                  4        /

$$\sqrt{2} \sqrt{\pi} \sqrt{x} \left(\frac{I_{- \frac{1}{2}}\left(x\right)}{4} + \frac{J_{- \frac{1}{2}}\left(x\right)}{4}\right)$$

sqrt(2)*sqrt(pi)*sqrt(x)*(besseli(-1/2, x)/4 + besselj(-1/2, x)/4)

Examples of finding the sum of a series

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```
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```
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```
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```
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```
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csc(n)^2/n^3
```
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1/n^2
```
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1/n^4
```
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1/n^6
```
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1/n
```
Grandi's series
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(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
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```
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```
Newton–Mercator series
```
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```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Identical expressions

Sum of series x^(4n)/(4n)!

The solution

Examples of finding the sum of a series