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n!/(7^n)
Sum of series/ n!/(7^n)

Sum of series n!/(7^n)


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The solution

You have entered [src] 
  oo    
____    
\   `   
 \    n!
  \   --
  /    n
 /    7 
/___,   
n = 1
$$\sum_{n=1}^{\infty} \frac{n!}{7^{n}}$$ 
Sum(factorial(n)/7^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n!}{7^{n}}$$
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
$$a_{n} = n!$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty} \left|{\frac{n!}{\left(n + 1\right)!}}\right|\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
  oo        
 ___        
 \  `       
  \    -n   
  /   7  *n!
 /__,       
n = 1
$$\sum_{n=1}^{\infty} 7^{- n} n!$$ 
Sum(7^(-n)*factorial(n), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series n!/(7^n)

    Examples of finding the sum of a series

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