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

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


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The solution

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

False
The rate of convergence of the power series
The graph
Sum of series sqrt(n+1)/(sqrt(n^2+n))

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