The geometric series which diverges is shown in the option number c as the absolute value of the common ratio of this series 4.
![\sum_{n=1 }^{\infty} \dfrac{2}{3}(-4)^{n-1}](https://tex.z-dn.net/?f=%5Csum_%7Bn%3D1%20%7D%5E%7B%5Cinfty%7D%20%5Cdfrac%7B2%7D%7B3%7D%28-4%29%5E%7Bn-1%7D)
<h3>What is geometric series diverges? </h3>
Geometric sequence is the sequence in which the next term is obtained by multiplying the previous term with the same number for the whole series.
It can be given as,
![a+ar+ ar^2+ ar^3+...](https://tex.z-dn.net/?f=a%2Bar%2B%20ar%5E2%2B%20ar%5E3%2B...)
Here,
is the (<em>a</em>) first term of the sequence, and (<em>r</em>) is the common ratio.
To be a series as geometric series diverges, it should follow,
![|r| > 1](https://tex.z-dn.net/?f=%7Cr%7C%20%3E%201)
First option given as,
3/5+3/10+3/20+3/40+ ...
Here, the common ratio is,
![r=\dfrac{\dfrac{3}{10}}{\dfrac{3}{5}}\\r=\dfrac{1}{2}](https://tex.z-dn.net/?f=r%3D%5Cdfrac%7B%5Cdfrac%7B3%7D%7B10%7D%7D%7B%5Cdfrac%7B3%7D%7B5%7D%7D%5C%5Cr%3D%5Cdfrac%7B1%7D%7B2%7D)
The common ratio is less than one. Thus option a is not correct.
First option given as,
-10+4-8/4+18/25- ...
Here, the common ratio is,
![r=\dfrac{4}{-10}\\r=\dfrac{-2}{5}](https://tex.z-dn.net/?f=r%3D%5Cdfrac%7B4%7D%7B-10%7D%5C%5Cr%3D%5Cdfrac%7B-2%7D%7B5%7D)
For the option number two the common ratio is (-2/5) which is less than 1. This option is also not correct.
In the option number c, the value of common ratio is -4. The absolute value of common ratio is,
![|r|=|-4|\\|r|=4](https://tex.z-dn.net/?f=%7Cr%7C%3D%7C-4%7C%5C%5C%7Cr%7C%3D4)
The absolute value of this common ratio is more than one. Thus, this is the correct option.
The geometric series which diverges is shown in the option number c as the absolute value of the common ratio of this series 4.
![\sum_{n=1 }^{\infty} \dfrac{2}{3}(-4)^{n-1}](https://tex.z-dn.net/?f=%5Csum_%7Bn%3D1%20%7D%5E%7B%5Cinfty%7D%20%5Cdfrac%7B2%7D%7B3%7D%28-4%29%5E%7Bn-1%7D)
Learn more about the geometric sequence here;
brainly.com/question/1509142