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Sum of series/ 1/(factorial(k+n)/factorial(n-1))

Sum of series 1/(factorial(k+n)/factorial(n-1))


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \        1     
  \   ----------
   )  /(k + n)!\
  /   |--------|
 /    \(n - 1)!/
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(k + n\right)! \frac{1}{\left(n - 1\right)!}}$$ 
Sum(1/(factorial(k + n)/factorial(n - 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(k + n\right)! \frac{1}{\left(n - 1\right)!}}$$
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
$$a_{n} = \frac{\left(n - 1\right)!}{\left(k + n\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n - 1\right)! \left(k + n + 1\right)!}{n! \left(k + n\right)!}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\left(n - 1\right)! \left(k + n + 1\right)!}{n! \left(k + n\right)!}}\right|$$
False
The answer [src] 
/    Gamma(k)                  
| -------------   for re(k) > 0
|      2                       
| Gamma (1 + k)                
|                              
|  oo                          
< ___                          
| \  `                         
|  \   (-1 + n)!               
|   )  ---------    otherwise  
|  /    (k + n)!               
| /__,                         
\n = 1
$$\begin{cases} \frac{\Gamma\left(k\right)}{\Gamma^{2}\left(k + 1\right)} & \text{for}\: \operatorname{re}{\left(k\right)} > 0 \\\sum_{n=1}^{\infty} \frac{\left(n - 1\right)!}{\left(k + n\right)!} & \text{otherwise} \end{cases}$$ 
Piecewise((gamma(k)/gamma(1 + k)^2, re(k) > 0), (Sum(factorial(-1 + n)/factorial(k + n), (n, 1, oo)), True))

    Examples of finding the sum of a series

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