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

Sum of series sin(pi/n)^n

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  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)

False

False

The rate of convergence of the power series

Numerical answer [src]

1.98873236754610398181308175190

1.98873236754610398181308175190

The graph