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(factorial(n-1)/factorial(n+1))^(n*(n+1))
Sum of series/ (factorial(n-1)/factorial(n+1))^(n*(n+1))

Sum of series (factorial(n-1)/factorial(n+1))^(n*(n+1))


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
0.0000214334706197012364032803137438
0.0000214334706197012364032803137438
The graph
Sum of series (factorial(n-1)/factorial(n+1))^(n*(n+1))

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