Mister Exam

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Sum of series factorial(n)/((n*n))

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  oo     
 ___     
 \  `    
  \    n!
   )  ---
  /   n*n
 /__,    
n = 1

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

Sum(factorial(n)/((n*n)), (n, 1, oo))

The radius of convergence of the power series

Given number:

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

and

x_{0} = 0

d = 0

c = 1

then

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

False

False

The rate of convergence of the power series

The answer [src]

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

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

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

Numerical answer

The series diverges

The graph