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1/((n(n+1)(n+2)(n+3)))

Sum of series 1/((n(n+1)(n+2)(n+3)))



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The solution

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  oo                           
 ___                           
 \  `                          
  \               1            
   )  -------------------------
  /   n*(n + 1)*(n + 2)*(n + 3)
 /__,                          
n = 1                          
$$\sum_{n=1}^{\infty} \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
Sum(1/(((n*(n + 1))*(n + 2))*(n + 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right) \left(n + 2\right) \left(n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n + 4}{n}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src]
1/18
$$\frac{1}{18}$$
1/18
Numerical answer [src]
0.0555555555555555555555555555556
0.0555555555555555555555555555556
The graph
Sum of series 1/((n(n+1)(n+2)(n+3)))

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