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sin((n+1)/(n^3+n-1))
Sum of series/ sin((n+1)/(n^3+n-1))

Sum of series sin((n+1)/(n^3+n-1))


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

False
The rate of convergence of the power series
The graph
Sum of series sin((n+1)/(n^3+n-1))

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