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(sin(pi/2n))^n
Sum of series/ (sin(pi/2n))^n

Sum of series (sin(pi/2n))^n


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The solution

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

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