Sum of series arctg(5/n)/n!

The solution

You have entered [src]

  oo         
____         
\   `        
 \        /5\
  \   atan|-|
   )      \n/
  /   -------
 /       n!  
/___,        
n = 1

$$\sum_{n=1}^{\infty} \frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$

Sum(atan(5/n)/factorial(n), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
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
$$a_{n} = \frac{\operatorname{atan}{\left(\frac{5}{n} \right)}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(n + 1\right)!}{n!}}\right| \operatorname{atan}{\left(\frac{5}{n} \right)}}{\operatorname{atan}{\left(\frac{5}{n + 1} \right)}}\right)$$
Let's take the limit
we find

False

False

The rate of convergence of the power series

Numerical answer [src]

2.18525920916049912044584755561

2.18525920916049912044584755561

The graph

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
```

Other calculators

How to use it?

Identical expressions

Sum of series arctg(5/n)/n!

The solution

Examples of finding the sum of a series