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Sum of series/ x^2(n+1)/n!

Sum of series x^2(n+1)/n!


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \     2        
  \   x *(n + 1)
  /   ----------
 /        n!    
/___,           
n = 0
$$\sum_{n=0}^{\infty} \frac{x^{2} \left(n + 1\right)}{n!}$$ 
Sum((x^2*(n + 1))/factorial(n), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{2} \left(n + 1\right)}{n!}$$
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
$$a_{n} = \frac{x^{2} \left(n + 1\right)}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{n + 2}\right)$$
Let's take the limit
we find
False

False
The answer [src] 
     2
2*E*x
$$2 e x^{2}$$ 
2*E*x^2

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