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factorial(n+1)/n^n

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



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The solution

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

False

False
The rate of convergence of the power series
The answer [src]
  oo              
 ___              
 \  `             
  \    -n         
  /   n  *(1 + n)!
 /__,             
n = 1             
$$\sum_{n=1}^{\infty} n^{- n} \left(n + 1\right)!$$
Sum(n^(-n)*factorial(1 + n), (n, 1, oo))
Numerical answer [src]
5.28287307046359212023905463722
5.28287307046359212023905463722
The graph
Sum of series factorial(n+1)/n^n
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