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1/((3n-1)(3n+2))

Sum of series 1/((3n-1)(3n+2))



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The solution

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  oo                     
 ___                     
 \  `                    
  \            1         
   )  -------------------
  /   (3*n - 1)*(3*n + 2)
 /__,                    
n = 1                    
n=11(3n1)(3n+2)\sum_{n=1}^{\infty} \frac{1}{\left(3 n - 1\right) \left(3 n + 2\right)}
Sum(1/((3*n - 1)*(3*n + 2)), (n, 1, oo))
The radius of convergence of the power series
Given number:
1(3n1)(3n+2)\frac{1}{\left(3 n - 1\right) \left(3 n + 2\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=1(3n1)(3n+2)a_{n} = \frac{1}{\left(3 n - 1\right) \left(3 n + 2\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((3n+5)13n1)1 = \lim_{n \to \infty}\left(\left(3 n + 5\right) \left|{\frac{1}{3 n - 1}}\right|\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.050.20
The answer [src]
  Gamma(8/3) 
-------------
10*Gamma(5/3)
Γ(83)10Γ(53)\frac{\Gamma\left(\frac{8}{3}\right)}{10 \Gamma\left(\frac{5}{3}\right)}
gamma(8/3)/(10*gamma(5/3))
Numerical answer [src]
0.166666666666666666666666666667
0.166666666666666666666666666667
The graph
Sum of series 1/((3n-1)(3n+2))

    Examples of finding the sum of a series