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

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



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The solution

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  oo                     
 ___                     
 \  `                    
  \            1         
   )  -------------------
  /   (2*n - 1)*(2*n + 3)
 /__,                    
n = 1                    
n=11(2n1)(2n+3)\sum_{n=1}^{\infty} \frac{1}{\left(2 n - 1\right) \left(2 n + 3\right)}
Sum(1/((2*n - 1)*(2*n + 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
1(2n1)(2n+3)\frac{1}{\left(2 n - 1\right) \left(2 n + 3\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(2n1)(2n+3)a_{n} = \frac{1}{\left(2 n - 1\right) \left(2 n + 3\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn((2n+1)(2n+5)12n12n+3)1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 1\right) \left(2 n + 5\right) \left|{\frac{1}{2 n - 1}}\right|}{2 n + 3}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.10.4
The answer [src]
1/3
13\frac{1}{3}
1/3
Numerical answer [src]
0.333333333333333333333333333333
0.333333333333333333333333333333
The graph
Sum of series 1/((2n-1)(2n+3))

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