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factorial(n+2)/n^n
Sum of series/ factorial(n+2)/n^n

Sum of series factorial(n+2)/n^n


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    (n + 2)!
  \   --------
  /       n   
 /       n    
/___,         
n = 1
∑n=1∞(n+2)!nn\sum_{n=1}^{\infty} \frac{\left(n + 2\right)!}{n^{n}} 
Sum(factorial(n + 2)/n^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
(n+2)!nn\frac{\left(n + 2\right)!}{n^{n}}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=n−n(n+2)!a_{n} = n^{- n} \left(n + 2\right)!
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(n−n(n+1)n+1∣(n+2)!(n+3)!∣)1 = \lim_{n \to \infty}\left(n^{- n} \left(n + 1\right)^{n + 1} \left|{\frac{\left(n + 2\right)!}{\left(n + 3\right)!}}\right|\right)
Let's take the limit
we find
False

False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5040
The answer [src] 
  oo              
 ___              
 \  `             
  \    -n         
  /   n  *(2 + n)!
 /__,             
n = 1
∑n=1∞n−n(n+2)!\sum_{n=1}^{\infty} n^{- n} \left(n + 2\right)! 
Sum(n^(-n)*factorial(2 + n), (n, 1, oo))
Numerical answer [src] 
22.5810160873374012485832230032
22.5810160873374012485832230032
The graph
Sum of series factorial(n+2)/n^n

    Examples of finding the sum of a series

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