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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
nan
$$\text{NaN}$$ 
nan
Numerical answer [src] 
0.193147180559945309417232121458
0.193147180559945309417232121458
The graph
Sum of series 1/((2n-1)(2n)(2n+1))

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