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factorial(n)/(n^3+n+8)
Sum of series/ factorial(n)/(n^3+n+8)

Sum of series factorial(n)/(n^3+n+8)


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The solution

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  oo            
____            
\   `           
 \        n!    
  \   ----------
  /    3        
 /    n  + n + 8
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{n!}{\left(n^{3} + n\right) + 8}$$ 
Sum(factorial(n)/(n^3 + n + 8), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n!}{\left(n^{3} + n\right) + 8}$$
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
$$a_{n} = \frac{n!}{n^{3} + n + 8}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + \left(n + 1\right)^{3} + 9\right) \left|{\frac{n!}{\left(n + 1\right)!}}\right|}{n^{3} + n + 8}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series factorial(n)/(n^3+n+8)

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