Question

Identify whether these series are divergent or convergent geometric series and find the sum, if possible.

Answer 1

A geometric series:

`\sum^(\infty)_(i=1)=a_1 * r^(i-1)`
It's convergent if |r|<1.
It's divergent if |r|≥1.
The sum can be found if it's a convergent series; it's equal to
`(a_1)/(1-r)`.

3.

`\sum^(\infty)_(i=1) 12 ((3)/(5))^(i-1) \\ \\ a_1=12 \\ r=(3)/(5) \\ \\ |r|<1 \hbox{ so it's convergent} \\ \\ \sum^(\infty)_(i=1) 12 ((3)/(5))^(i-1)=(12)/(1-(3)/(5))=(12)/((5)/(5)-(3)/(5))=(12)/((2)/(5))=12 * (5)/(2)=6 * 5=30`

The answer is: This is a convergent geometric series. The sum is 30.

4.

`\sum^(\infty)_(i=1) 15(4)^(i-1) \\ \\ a_1=15 \\ r=4 \\ \\ |r| \geq 1 \hbox{ so it's divergent}`

The answer is: This is a divergent geometric series. The sum cannot be found.