Sum of series n^2/n!

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  oo    
____    
\   `   
 \     2
  \   n 
  /   --
 /    n!
/___,   
n = 1

\sum_{n=1}^{\infty} \frac{n^{2}}{n!}

Sum(n^2/factorial(n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{n^{2}}{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^{2}}{n!}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{n^{2} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{\left(n + 1\right)^{2}}\right)

False

False

The rate of convergence of the power series

The answer [src]

2*E

2 e

2*E

Numerical answer [src]

5.43656365691809047072057494271

5.43656365691809047072057494271

The graph