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arcsin(1/n)/(sqrt(n+1)-sqrt(n))
Sum of series/ arcsin(1/n)/(sqrt(n+1)-sqrt(n))

Sum of series arcsin(1/n)/(sqrt(n+1)-sqrt(n))


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The solution

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

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

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