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((3+sin(n))/(sqrt((n^3)-n)))
Sum of series/ ((3+sin(n))/(sqrt((n^3)-n)))

Sum of series ((3+sin(n))/(sqrt((n^3)-n)))


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  oo             
____             
\   `            
 \     3 + sin(n)
  \   -----------
   )     ________
  /     /  3     
 /    \/  n  - n 
/___,            
n = 2
$$\sum_{n=2}^{\infty} \frac{\sin{\left(n \right)} + 3}{\sqrt{n^{3} - n}}$$ 
Sum((3 + sin(n))/sqrt(n^3 - n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(n \right)} + 3}{\sqrt{n^{3} - n}}$$
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
$$a_{n} = \frac{\sin{\left(n \right)} + 3}{\sqrt{n^{3} - n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(\sin{\left(n \right)} + 3\right) \sqrt{- n + \left(n + 1\right)^{3} - 1}}{\sin{\left(n + 1 \right)} + 3}}\right|}{\left|{\sqrt{n^{3} - n}}\right|}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{\left(\sin{\left(n \right)} + 3\right) \sqrt{- n + \left(n + 1\right)^{3} - 1}}{\sin{\left(n + 1 \right)} + 3}}\right|}{\left|{\sqrt{n^{3} - n}}\right|}\right)$$
False
The rate of convergence of the power series
The graph
Sum of series ((3+sin(n))/(sqrt((n^3)-n)))

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