Answer:
The correct option is C.
Step-by-step explanation:
A geometric series divergent if [tex]|r|\geq1[/tex].
In the first option the first term of the series is,
[tex]a=\frac{2}{5}[/tex]
common ratio is
[tex]r=\frac{3/10}{3/5} =\frac{1}{2}[/tex]
Since the common ratio is less than 1, therefore the geometric series is convergent and the option A is incorrect.
In the second option the first term of the series is,
[tex]a=-10[/tex]
common ratio is
[tex]r=\frac{4}{-10} =-\frac{2}{5}[/tex]
Since the common ratio is less than 1, therefore the geometric series is convergent and the option B is incorrect.
The nth term of a geometric series is in the form of
[tex]a_n=ar^{n-1}[/tex]
So, the common ratio of option C and D are -4 and [tex]\frac{1}{5}[/tex] respectively.
Since the absolute common ratio in option C is more than 1. i.e., [tex]|-4|\geq1[/tex], therefore the geometric series is divergent and the option C is correct.
Since the common ratio in option D is less than 1, therefore the geometric series is convergent and the option D is incorrect.
