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2*n/(1+n^2)

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  • Sum of series:
  • (n^2-3n-18) (n^2-3n-18)
  • 2^(k+3)/5^(k+1) 2^(k+3)/5^(k+1)
  • 2*n/(1+n^2) 2*n/(1+n^2)
  • 1/9n^2-3n-2 1/9n^2-3n-2
  • Limit of the function:
  • 2*n/(1+n^2) 2*n/(1+n^2)

  • Identical expressions

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

  • Similar expressions

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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo        
____        
\   `       
 \     2*n  
  \   ------
  /        2
 /    1 + n 
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{2 n}{n^{2} + 1}$$ 
Sum(2*n/(1 + n^2), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series 2*n/(1+n^2)

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