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1/(2x+1)^2
Sum of series/ 1/(2x+1)^2

Sum of series 1/(2x+1)^2


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \        1     
  \   ----------
  /            2
 /    (2*x + 1) 
/___,           
x = 1
$$\sum_{x=1}^{\infty} \frac{1}{\left(2 x + 1\right)^{2}}$$ 
Sum(1/((2*x + 1)^2), (x, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(2 x + 1\right)^{2}}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{\left(2 x + 1\right)^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{\left(2 x + 3\right)^{2}}{\left(2 x + 1\right)^{2}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
       2
     pi 
-1 + ---
      8
$$-1 + \frac{\pi^{2}}{8}$$ 
-1 + pi^2/8
Numerical answer [src] 
0.233700550136169827354311374985
0.233700550136169827354311374985
The graph
Sum of series 1/(2x+1)^2

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