Sum of series pi^n/n!

The solution

You have entered [src]

  oo     
____     
\   `    
 \      n
  \   pi 
  /   ---
 /     n!
/___,    
n = 1

$$\sum_{n=1}^{\infty} \frac{\pi^{n}}{n!}$$

Sum(pi^n/factorial(n), (n, 1, oo))

The radius of convergence of the power series

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

The rate of convergence of the power series

The answer [src]

   /        pi\
   |  1    e  |
pi*|- -- + ---|
   \  pi    pi/

$$\pi \left(- \frac{1}{\pi} + \frac{e^{\pi}}{\pi}\right)$$

pi*(-1/pi + exp(pi)/pi)

Numerical answer [src]

22.1406926327792690057290863679

22.1406926327792690057290863679

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
```

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Identical expressions

Sum of series pi^n/n!

The solution

Examples of finding the sum of a series