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1/((2n+1)(2n+5))
Sum of series/ 1/((2n+1)(2n+5))

Sum of series 1/((2n+1)(2n+5))


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The solution

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  oo                     
 ___                     
 \  `                    
  \            1         
   )  -------------------
  /   (2*n + 1)*(2*n + 5)
 /__,                    
n = 1
∑n=1∞1(2n+1)(2n+5)\sum_{n=1}^{\infty} \frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)} 
Sum(1/((2*n + 1)*(2*n + 5)), (n, 1, oo))
The radius of convergence of the power series
Given number:
1(2n+1)(2n+5)\frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)}
It is a series of species
an(cx−x0)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+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=1(2n+1)(2n+5)a_{n} = \frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((2n+3)(2n+7)(2n+1)(2n+5))1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 3\right) \left(2 n + 7\right)}{\left(2 n + 1\right) \left(2 n + 5\right)}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.000.15
The answer [src] 
2/15
215\frac{2}{15} 
2/15
Numerical answer [src] 
0.133333333333333333333333333333
0.133333333333333333333333333333
The graph
Sum of series 1/((2n+1)(2n+5))

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