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factorial(n+1)/8^(n+1)
Sum of series/ factorial(n+1)/8^(n+1)

Sum of series factorial(n+1)/8^(n+1)


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The solution

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  oo          
____          
\   `         
 \    (n + 1)!
  \   --------
  /     n + 1 
 /     8      
/___,         
n = 1
∑n=1∞(n+1)!8n+1\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{8^{n + 1}} 
Sum(factorial(n + 1)/8^(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
(n+1)!8n+1\frac{\left(n + 1\right)!}{8^{n + 1}}
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=8−n−1(n+1)!a_{n} = 8^{- n - 1} \left(n + 1\right)!
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(8−n−1⋅8n+2∣(n+1)!(n+2)!∣)1 = \lim_{n \to \infty}\left(8^{- n - 1} \cdot 8^{n + 2} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)
Let's take the limit
we find
False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.000.10
The answer [src] 
  oo                  
 ___                  
 \  `                 
  \    -1 - n         
  /   8      *(1 + n)!
 /__,                 
n = 1
∑n=1∞8−n−1(n+1)!\sum_{n=1}^{\infty} 8^{- n - 1} \left(n + 1\right)! 
Sum(8^(-1 - n)*factorial(1 + n), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series factorial(n+1)/8^(n+1)

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