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

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


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The solution

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  oo                               
 ___                               
 \  `                              
  \                 1              
   )  -----------------------------
  /   (2*n - 1)*(2*n + 1)*(2*n + 3)
 /__,                              
n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right) \left(2 n + 3\right)}$$ 
Sum(1/(((2*n - 1)*(2*n + 1))*(2*n + 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(2 n - 1\right) \left(2 n + 1\right) \left(2 n + 3\right)}$$
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
$$a_{n} = \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right) \left(2 n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(2 n + 5\right) \left|{\frac{1}{2 n - 1}}\right|\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
1/12
$$\frac{1}{12}$$ 
1/12
Numerical answer [src] 
0.0833333333333333333333333333333
0.0833333333333333333333333333333
The graph
Sum of series 1/((2n-1)(2n+1)(2n+3))

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