Question

How do you find the sum of the infinite geometric series 7/8 + 7/12 + 7/18 + 7/27 ...?

Answer 1

This sum exists and is equal to `21/8`.

Explanation:

General form of geometric series:
`a+aq+aq^2+aq^3+...`

Factor out first term `a` to see what the quotient `q` of this geometric series could be:

`7/8(1+8/7*7/12+8/7*7/18+8/7*7/27+...)=7/8(1+2/3+4/9+8/27+...)=7/8(1+2/3+(2/3)^2+(2/3)^3+...)`

Now we could see that `a=7/8` and `q=2/3`. As long as `-1< q<1` the series converges and sum exists:
`a+aq+aq^2+aq^3+...=a/(1-q)=(7/8)/(1-2/3)=7/8*3=21/8`