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

The solution

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  oo                     
 ___                     
 \  `                    
  \            1         
   )  -------------------
  /   (2*n + 1)*(2*n + 5)
 /__,                    
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)}$$

Sum(1/((2*n + 1)*(2*n + 5)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)}$$
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
$$a_{n} = \frac{1}{\left(2 n + 1\right) \left(2 n + 5\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 3\right) \left(2 n + 7\right)}{\left(2 n + 1\right) \left(2 n + 5\right)}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

2/15

$$\frac{2}{15}$$

2/15

Numerical answer [src]

0.133333333333333333333333333333

0.133333333333333333333333333333

The graph

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```
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Exam the series for convergence
```
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Sum of series 1/((2n+1)(2n+5))

The solution

Examples of finding the sum of a series