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

The solution

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  oo         
 ___         
 \  `        
  \   3*n - 1
   )  -------
  /   3*n + 2
 /__,        
n = 1

$$\sum_{n=1}^{\infty} \frac{3 n - 1}{3 n + 2}$$

Sum((3*n - 1)/(3*n + 2), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{3 n - 1}{3 n + 2}$$
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
$$a_{n} = \frac{3 n - 1}{3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 5\right) \left|{3 n - 1}\right|}{\left(3 n + 2\right)^{2}}\right)$$
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

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```
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```
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```
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Newton–Mercator series
```
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```
Exam the series for convergence
```
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```

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

The solution

Examples of finding the sum of a series