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(n-1)/(n(n+1)(n+2))
Sum of series/ (n-1)/(n(n+1)(n+2))

Sum of series (n-1)/(n(n+1)(n+2))


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
1/4
$$\frac{1}{4}$$ 
1/4
Numerical answer [src] 
0.250000000000000000000000000000
0.250000000000000000000000000000
The graph
Sum of series (n-1)/(n(n+1)(n+2))

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