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sin(pi/n)^n
Sum of series/ sin(pi/n)^n

Sum of series sin(pi/n)^n


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
1.98873236754610398181308175190
1.98873236754610398181308175190
The graph
Sum of series sin(pi/n)^n

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