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arctg/n^2

  • How to use it?

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

  • Identical expressions

  • arctg/n^ two
  • arctg divide by n squared
  • arctg divide by n to the power of two
  • arctg/n2
  • arctg/n²
  • arctg/n to the power of 2
  • arctg divide by n^2
Sum of series/ arctg/n^2

Sum of series arctg/n^2


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.60728194331664439128535045493
1.60728194331664439128535045493
The graph
Sum of series arctg/n^2

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