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

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

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  oo        
____        
\   `       
 \    1 + n 
  \   ------
  /        2
 /    1 + n 
/___,       
n = 1

\sum_{n=1}^{\infty} \frac{n + 1}{n^{2} + 1}

Sum((1 + n)/(1 + n^2), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{n + 1}{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{n + 1}{n^{2} + 1}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(\left(n + 1\right)^{2} + 1\right)}{\left(n + 2\right) \left(n^{2} + 1\right)}\right)

True

False

The rate of convergence of the power series

Numerical answer

The series diverges

The graph