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arcsin^n(2/(n+1))

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  • Sum of series:
  • 1/n! 1/n!
  • 1/(2^n) 1/(2^n)
  • 2/n 2/n
  • cos(n)*x/n^2

  • Identical expressions

  • arcsin^n(two /(n+ one))
  • arc sinus of to the power of n(2 divide by (n plus 1))
  • arc sinus of to the power of n(two divide by (n plus one))
  • arcsinn(2/(n+1))
  • arcsinn2/n+1
  • arcsin^n2/n+1
  • arcsin^n(2 divide by (n+1))

  • Similar expressions

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

Sum of series arcsin^n(2/(n+1))


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
2.28072160680856057406320072294
2.28072160680856057406320072294
The graph
Sum of series arcsin^n(2/(n+1))

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