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factorial(n+1)/3^n

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



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The solution

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  oo          
____          
\   `         
 \    (n + 1)!
  \   --------
  /       n   
 /       3    
/___,         
n = 1         
$$\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{3^{n}}$$
Sum(factorial(n + 1)/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(n + 1\right)!}{3^{n}}$$
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
$$a_{n} = \left(n + 1\right)!$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src]
  oo              
 ___              
 \  `             
  \    -n         
  /   3  *(1 + n)!
 /__,             
n = 1             
$$\sum_{n=1}^{\infty} 3^{- n} \left(n + 1\right)!$$
Sum(3^(-n)*factorial(1 + n), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series factorial(n+1)/3^n
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