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sin^n(7pi/6)
Sum of series/ sin^n(7pi/6)

Sum of series sin^n(7pi/6)


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The solution

You have entered [src] 
  oo            
 ___            
 \  `           
  \      n/7*pi\
   )  sin |----|
  /       \ 6  /
 /__,           
n = 1
$$\sum_{n=1}^{\infty} \sin^{n}{\left(\frac{7 \pi}{6} \right)}$$ 
Sum(sin((7*pi)/6)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin^{n}{\left(\frac{7 \pi}{6} \right)}$$
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
$$a_{n} = 1$$
and
$$x_{0} = \frac{1}{2}$$
,
$$d = 1$$
,
$$c = 0$$
then
False

Let's take the limit
we find
False
The rate of convergence of the power series
The answer [src] 
-1/3
$$- \frac{1}{3}$$ 
-1/3
Numerical answer [src] 
-0.333333333333333333333333333333
-0.333333333333333333333333333333
The graph
Sum of series sin^n(7pi/6)

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