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6/((3n-1)*(3n+5))
Sum of series/ 6/((3n-1)*(3n+5))

Sum of series 6/((3n-1)*(3n+5))


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The solution

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  oo                     
 ___                     
 \  `                    
  \            6         
   )  -------------------
  /   (3*n - 1)*(3*n + 5)
 /__,                    
n = 1
$$\sum_{n=1}^{\infty} \frac{6}{\left(3 n - 1\right) \left(3 n + 5\right)}$$ 
Sum(6/(((3*n - 1)*(3*n + 5))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{6}{\left(3 n - 1\right) \left(3 n + 5\right)}$$
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
$$a_{n} = \frac{6}{\left(3 n - 1\right) \left(3 n + 5\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 2\right) \left(3 n + 8\right) \left|{\frac{1}{3 n - 1}}\right|}{3 n + 5}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
21*Gamma(11/3)
--------------
80*Gamma(8/3)
$$\frac{21 \Gamma\left(\frac{11}{3}\right)}{80 \Gamma\left(\frac{8}{3}\right)}$$ 
21*gamma(11/3)/(80*gamma(8/3))
Numerical answer [src] 
0.700000000000000000000000000000
0.700000000000000000000000000000
The graph
Sum of series 6/((3n-1)*(3n+5))

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