Mister Exam

Sum of series sin(3n)



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The solution

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

    Examples of finding the sum of a series