Mister Exam

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

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

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

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

The radius of convergence of the power series

Given number:

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

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

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

The answer [src]

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

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

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

The graph