Mister Exam

Other calculators

arctg(n)/(1+n^2)

  • How to use it?

  • Sum of series:
  • 12/(n^2+4*n+3) 12/(n^2+4*n+3)
  • 3/((n+2)^2-1) 3/((n+2)^2-1)
  • 2/(x^2+2x)
  • tan(1/2^n)/2^n tan(1/2^n)/2^n

  • Identical expressions

  • arctg(n)/(one +n^ two)
  • arctg(n) divide by (1 plus n squared )
  • arctg(n) divide by (one plus n to the power of two)
  • arctg(n)/(1+n2)
  • arctgn/1+n2
  • arctg(n)/(1+n²)
  • arctg(n)/(1+n to the power of 2)
  • arctgn/1+n^2
  • arctg(n) divide by (1+n^2)

  • Similar expressions

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

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


∑
=

The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.13558057875099994273174426656
1.13558057875099994273174426656
The graph
Sum of series arctg(n)/(1+n^2)

    Examples of finding the sum of a series

    © Mister Exam – Calculator