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factorial(n+1)/n^3

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  • factorial(n+ one)/n^ three
  • factorial(n plus 1) divide by n cubed
  • factorial(n plus one) divide by n to the power of three
  • factorial(n+1)/n3
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  • factorial(n-1)/n^3
Sum of series/ factorial(n+1)/n^3

Sum of series factorial(n+1)/n^3


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    (n + 1)!
  \   --------
  /       3   
 /       n    
/___,         
n = 1
∑n=1∞(n+1)!n3\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{n^{3}} 
Sum(factorial(n + 1)/n^3, (n, 1, oo))
The radius of convergence of the power series
Given number:
(n+1)!n3\frac{\left(n + 1\right)!}{n^{3}}
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+1)!n3a_{n} = \frac{\left(n + 1\right)!}{n^{3}}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((n+1)3∣(n+1)!(n+2)!∣n3)1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{3} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|}{n^{3}}\right)
Let's take the limit
we find
False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50200
Numerical answer
The series diverges
The graph
Sum of series factorial(n+1)/n^3

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