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1^n/(1+n^2)

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  • Sum of series:
  • x^(2*n)
  • 1^n/(1+n^2) 1^n/(1+n^2)
  • sqrt(9)*3 sqrt(9)*3
  • sen(5n)/k^2

  • Identical expressions

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

  • Similar expressions

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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo        
____        
\   `       
 \      1   
  \   ------
  /        2
 /    1 + n 
/___,       
n = 0
$$\sum_{n=0}^{\infty} \frac{1}{n^{2} + 1}$$ 
Sum(1/(1 + n^2), (n, 0, oo))
Numerical answer [src] 
2.07667404746858117413405079475
2.07667404746858117413405079475
The graph
Sum of series 1^n/(1+n^2)

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