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

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


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

False
The rate of convergence of the power series
The answer [src] 
  2
pi 
---
 8
$$\frac{\pi^{2}}{8}$$ 
pi^2/8
Numerical answer [src] 
1.23370055013616982735431137498
1.23370055013616982735431137498
The graph
Sum of series 1/((2n+1)(2n+1))

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