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Sum of series/ arcsin1/sqrt(n)

Sum of series arcsin1/sqrt(n)

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  oo         
____         
\   `        
 \    asin(1)
  \   -------
  /      ___ 
 /     \/ n  
/___,        
n = 1

\sum_{n=1}^{\infty} \frac{\operatorname{asin}{\left(1 \right)}}{\sqrt{n}}

Sum(asin(1)/sqrt(n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\operatorname{asin}{\left(1 \right)}}{\sqrt{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{\pi}{2 \sqrt{n}}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

The answer [src]

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

\sum_{n=1}^{\infty} \frac{\pi}{2 \sqrt{n}}

Sum(pi/(2*sqrt(n)), (n, 1, oo))

The graph