Question

How do you find the sum of the infinite geometric series 8 + 4 + 2 + 1 +...?

Answer 1

`16`

Explanation:

This converging geometric progression is a special one.
`"G.P " 8,4,2,1,1/2,1/4...`
Common multiple is `1/2` .

When you find the sum of this G.P
`=>8+4+2+1+1/2+1/4....`

`=>15+1/2+1/4+1/8.....`

`=>15+1`

Basically, you need to prove `1/2+1/4+1/8...=1`

Proof:
`S_n=1/2+1/4+...1/2^(n-1)`

Multiply `2` both sides

`2S_n=2/2+2/4+...1/2^n`

Subtract `S_n` from this.

`2S_n-S_n=1+2/4-1/2+2/8-1/4....`

`S_n=1-1/2^n`

As you increase the value of `n` the value of `S_n` tends to be `1`.

There's one more proof which is easy to understand and really amazing.

Given below is a square of side 1unit. First cut it half, then `(1/4 )^(th)` and then `(1/8) ^(th)` and so on. You'll see that as you increase the value of `n` the sum tends to be `1`.