Question

4−1+1/4−1/16+... Find the sum of the infinite geometric series, if it exists.

Answer 1

The sum is
`S=(16)/(5)=3.2`

Explanation:

To find the sum of the infinite geometric series we must first find the common ratio r.

The series is:

4-1 + 1 / 4-1 / 16 +

`r =(a_(n+1))/(a_n)`

`r=(-1)/(4)=-(1)/(4)\\\\r=((1)/(4))/(-1)=-(1)/(4)`

Then the common ratio r is

`r=-(1)/(4)`

The first term is:
`a_1=4`

By definition when
`0 <| r | <1` then the sum of the infinite sequence is:

`S=(a_1)/(1-r)\\\\S=(4)/(1-(-(1)/(4)))\\\\S=(4)/((5)/(4))\\\\S=(16)/(5)`