Sum of series 6/(3n-1)(3n+5)

The solution

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  oo                   
 ___                   
 \  `                  
  \      6             
   )  -------*(3*n + 5)
  /   3*n - 1          
 /__,                  
n = 1

$$\sum_{n=1}^{\infty} \frac{6}{3 n - 1} \left(3 n + 5\right)$$

Sum((6/(3*n - 1))*(3*n + 5), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{6}{3 n - 1} \left(3 n + 5\right)$$
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
$$a_{n} = \frac{6 \left(3 n + 5\right)}{3 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 2\right) \left(3 n + 5\right) \left|{\frac{1}{3 n - 1}}\right|}{3 n + 8}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

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$$\infty$$

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Numerical answer

The series diverges

The graph

Examples of finding the sum of a series

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1/2^(n!)
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k!/(n!*(n+k)!)
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Flint Hills Series
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csc(n)^2/n^3
```
Basel problem
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1/n^2
```
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1/n^4
```
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1/n^6
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Harmonic series
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1/n
```
Grandi's series
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(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Sum of series 6/(3n-1)(3n+5)

The solution

Examples of finding the sum of a series