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Sum of series/ n(a*x)^n/a*n!

Sum of series n(a*x)^n/a*n!


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The solution

You have entered [src] 
  oo             
____             
\   `            
 \           n   
  \   n*(a*x)    
  /   --------*n!
 /       a       
/___,            
n = 0
$$\sum_{n=0}^{\infty} \frac{n \left(a x\right)^{n}}{a} n!$$ 
Sum(((n*(a*x)^n)/a)*factorial(n), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{n \left(a x\right)^{n}}{a} n!$$
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
$$a_{n} = \frac{n n!}{a}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = a$$
then
$$R = \frac{\lim_{n \to \infty}\left(\frac{n \left|{\frac{n!}{\left(n + 1\right)!}}\right|}{n + 1}\right)}{a}$$
Let's take the limit
we find
$$R = 0$$
The answer [src] 
  oo             
____             
\   `            
 \           n   
  \   n*(a*x) *n!
  /   -----------
 /         a     
/___,            
n = 0
$$\sum_{n=0}^{\infty} \frac{n \left(a x\right)^{n} n!}{a}$$ 
Sum(n*(a*x)^n*factorial(n)/a, (n, 0, oo))

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