Sum of series factorial(x-n)

The solution

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  oo          
 __           
 \ `          
  )   (x - n)!
 /_,          
n = 1

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

Sum(factorial(x - n), (n, 1, oo))

The radius of convergence of the power series

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

True

False

The answer [src]

    -1  pi*I*x                    /   pi*I   pi*I\
-x*e  *e      *Gamma(x)*lowergamma\x*e    , e    /

$$- \frac{x e^{i \pi x} \Gamma\left(x\right) \gamma\left(x e^{i \pi}, e^{i \pi}\right)}{e}$$

-x*exp(-1)*exp(pi*i*x)*gamma(x)*lowergamma(x*exp_polar(pi*i), exp_polar(pi*i))

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

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Sum of series factorial(x-n)

The solution

Examples of finding the sum of a series