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sin(pi*n/3)
Sum of series/ sin(pi*n/3)

Sum of series sin(pi*n/3)


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The solution

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  oo           
 ___           
 \  `          
  \      /pi*n\
   )  sin|----|
  /      \ 3  /
 /__,          
n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{3} \right)}$$ 
Sum(sin((pi*n)/3), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin{\left(\frac{\pi n}{3} \right)}$$
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
$$a_{n} = \sin{\left(\frac{\pi n}{3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{\pi n}{3} \right)}}{\sin{\left(\pi \left(\frac{n}{3} + \frac{1}{3}\right) \right)}}}\right|$$
False
The rate of convergence of the power series
The answer [src] 
  oo           
 ___           
 \  `          
  \      /pi*n\
   )  sin|----|
  /      \ 3  /
 /__,          
n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi n}{3} \right)}$$ 
Sum(sin(pi*n/3), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series sin(pi*n/3)

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