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

1 Answer
Mar 3, 2018

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.

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