Question

Does this Series Converge or Diverge? What is the common ratio?

Answer 1

Converge

Explanation:

Given geometric series

`1,000+200+40+8+(8)/(5)+...`

The first term of this geometric series is
`a_1=1,000.`

Next terms can be obtained by dividing by 5 or multiplying by
`(1)/(5):`

`a_2=1,000\cdot (1)/(5)=200\\ \\a_3=200\cdot (1)/(5)=40\\ \\a_4=40\cdot (1)/(5)=8\\ \\a_5=8\cdot (1)/(5)=(8)/(5)\\ \\...`

The sum of this infinite geometric series is

`S=(a_1)/(1-q)=(1,000)/(1-(1)/(5))=(1,000)/((4)/(5))=1,250`

Hence, this series is convergent