Which geometric series diverges? Three-fifths + three-tenths + three-twentieths + StartFraction 3 Over 40 EndFraction + ellipsis Negative 10 + 4 minus eight-fifths + StartFraction 16 Over 25 EndFraction minus ellipsis Sigma-Summation Underscript n = 1 Overscript infinity EndScripts two-thirds (negative 4) Superscript n minus 1 Sigma-Summation Underscript n = 1 Overscript infinity EndScripts (negative 12) (one-fifth) Superscript n minus 1

Options:

a. Three-fifths + three-tenths + three-twentieths + StartFraction 3 Over 40 EndFraction + ellipsis 

b. Negative 10 + 4 minus eight-fifths + StartFraction 16 Over 25 EndFraction minus ellipsis 

c. Sigma-Summation Underscript n = 1 Overscript infinity EndScripts two-thirds (negative 4) Superscript n minus 1 

d. Sigma-Summation Underscript n = 1 Overscript infinity EndScripts (negative 12) (one-fifth) Superscript n minus 1

Answer:

c. Sigma-Summation Underscript n = 1 Overscript infinity EndScripts two-thirds (negative 4) Superscript n minus 1 

Step-by-step explanation:

A geometric series divergent if .

In the first option the first term of the series is,

common ratio is

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,

common ratio is

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

So, the common ratio of option C and D are -4 and respectively.

Since the absolute common ratio in option C is more than 1. i.e., , 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.