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

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


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The solution

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  oo                            
_____                           
\    `                          
 \     /                 ___   \
  \    |  _______      \/ n    |
   \   |\/ n + 1  - -----------|
   /   |               ________|
  /    |              /  2     |
 /     \            \/  n  + n /
/____,                          
n = 1
$$\sum_{n=1}^{\infty} \left(- \frac{\sqrt{n}}{\sqrt{n^{2} + n}} + \sqrt{n + 1}\right)$$ 
Sum(sqrt(n + 1) - sqrt(n)/sqrt(n^2 + n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{\sqrt{n}}{\sqrt{n^{2} + n}} + \sqrt{n + 1}$$
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
$$a_{n} = - \frac{\sqrt{n}}{\sqrt{n^{2} + n}} + \sqrt{n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\sqrt{n}}{\sqrt{n^{2} + n}} - \sqrt{n + 1}}{\frac{\sqrt{n + 1}}{\sqrt{n + \left(n + 1\right)^{2} + 1}} - \sqrt{n + 2}}}\right|$$
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)/sqrt(n^2+n)

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