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6/(9n^2+6n-8)
Sum of series/ 6/(9n^2+6n-8)

Sum of series 6/(9n^2+6n-8)


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The solution

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  oo                
____                
\   `               
 \          6       
  \   --------------
  /      2          
 /    9*n  + 6*n - 8
/___,               
n = 1
$$\sum_{n=1}^{\infty} \frac{6}{\left(9 n^{2} + 6 n\right) - 8}$$ 
Sum(6/(9*n^2 + 6*n - 8), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{6}{\left(9 n^{2} + 6 n\right) - 8}$$
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
$$a_{n} = \frac{6}{9 n^{2} + 6 n - 8}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(6 n + 9 \left(n + 1\right)^{2} - 2\right) \left|{\frac{1}{9 n^{2} + 6 n - 8}}\right|\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
15*Gamma(10/3)
--------------
28*Gamma(7/3)
$$\frac{15 \Gamma\left(\frac{10}{3}\right)}{28 \Gamma\left(\frac{7}{3}\right)}$$ 
15*gamma(10/3)/(28*gamma(7/3))
Numerical answer [src] 
1.25000000000000000000000000000
1.25000000000000000000000000000
The graph
Sum of series 6/(9n^2+6n-8)

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