Mister Exam

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

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

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

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

The radius of convergence of the power series

Given number:

\frac{n!}{n + 1}

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 + 1}

and

x_{0} = 0

d = 0

c = 1

then

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

False

False

The rate of convergence of the power series

Numerical answer

The series diverges

The graph