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Sum of series/ narcsin3n/2n^3+1

Sum of series narcsin3n/2n^3+1


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  oo                      
 ___                      
 \  `                     
  \   /n*asin(3*n)  3    \
   )  |-----------*n  + 1|
  /   \     2            /
 /__,                     
n = 3
$$\sum_{n=3}^{\infty} \left(n^{3} \frac{n \operatorname{asin}{\left(3 n \right)}}{2} + 1\right)$$ 
Sum(((n*asin(3*n))/2)*n^3 + 1, (n, 3, oo))
The radius of convergence of the power series
Given number:
$$n^{3} \frac{n \operatorname{asin}{\left(3 n \right)}}{2} + 1$$
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
$$a_{n} = \frac{n^{4} \operatorname{asin}{\left(3 n \right)}}{2} + 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{n^{4} \operatorname{asin}{\left(3 n \right)}}{2} + 1}{\frac{\left(n + 1\right)^{4} \operatorname{asin}{\left(3 n + 3 \right)}}{2} + 1}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{n^{4} \operatorname{asin}{\left(3 n \right)}}{2} + 1}{\frac{\left(n + 1\right)^{4} \operatorname{asin}{\left(3 n + 3 \right)}}{2} + 1}}\right|$$
False
The answer [src] 
  oo                    
____                    
\   `                   
 \    /     4          \
  \   |    n *asin(3*n)|
  /   |1 + ------------|
 /    \         2      /
/___,                   
n = 3
$$\sum_{n=3}^{\infty} \left(\frac{n^{4} \operatorname{asin}{\left(3 n \right)}}{2} + 1\right)$$ 
Sum(1 + n^4*asin(3*n)/2, (n, 3, oo))

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