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arcsin(1/n)^n

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  • Sum of series:
  • (3/4)^n (3/4)^n
  • 1/((2n-1)*(2n+1)) 1/((2n-1)*(2n+1))
  • 1/(n*(n+3)) 1/(n*(n+3))
  • (5/6)^n (5/6)^n

  • Identical expressions

  • arcsin(one /n)^n
  • arc sinus of (1 divide by n) to the power of n
  • arc sinus of (one divide by n) to the power of n
  • arcsin(1/n)n
  • arcsin1/nn
  • arcsin1/n^n
  • arcsin(1 divide by n)^n
Sum of series/ arcsin(1/n)^n

Sum of series arcsin(1/n)^n


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.88863031385355231616976511810
1.88863031385355231616976511810
The graph
Sum of series arcsin(1/n)^n

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