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

The solution

You have entered [src]

  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]

oo

\infty

oo

Numerical answer

The series diverges

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-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
```

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series