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arctg(n)/(2*n^2-1)

  • How to use it?

  • Sum of series:
  • 1/((2n-1)(2n+1)) 1/((2n-1)(2n+1))
  • (n+2/2n)^n2
  • x/(x-1) x/(x-1)
  • sin(n*x)/5^n

  • Identical expressions

  • arctg(n)/(two *n^ two - one)
  • arctg(n) divide by (2 multiply by n squared minus 1)
  • arctg(n) divide by (two multiply by n to the power of two minus one)
  • arctg(n)/(2*n2-1)
  • arctgn/2*n2-1
  • arctg(n)/(2*n²-1)
  • arctg(n)/(2*n to the power of 2-1)
  • arctg(n)/(2n^2-1)
  • arctg(n)/(2n2-1)
  • arctgn/2n2-1
  • arctgn/2n^2-1
  • arctg(n) divide by (2*n^2-1)

  • Similar expressions

  • arctg(n)/(2*n^2+1)
Sum of series/ arctg(n)/(2*n^2-1)

Sum of series arctg(n)/(2*n^2-1)


∑
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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.22280447076102925481332706415
1.22280447076102925481332706415
The graph
Sum of series arctg(n)/(2*n^2-1)

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