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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                    
n=16(3n1)(3n+5)\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:
6(3n1)(3n+5)\frac{6}{\left(3 n - 1\right) \left(3 n + 5\right)}
It is a series of species
an(cxx0)dna_{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:
Rd=x0+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=6(3n1)(3n+5)a_{n} = \frac{6}{\left(3 n - 1\right) \left(3 n + 5\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((3n+2)(3n+8)13n13n+5)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
1.07.01.52.02.53.03.54.04.55.05.56.06.50.250.75
The answer [src]
21*Gamma(11/3)
--------------
80*Gamma(8/3) 
21Γ(113)80Γ(83)\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))

    Examples of finding the sum of a series