Half, half, half

There are various ways to determine the sum of 1/2 + 1/4 + 1/8 + ..., but here is a simple way:

Let s be the sum of the series s = 1/2 + 1/4 + 1/8 + ...

Let a be the sum of this similar series a = 1/4 + 1/8 + 1/16 ...

By multiplying the second series by 2 it is clear that 2a = s

And by subtracting the second series from the first it is clear that s-a = 1/2

From the last equality we get a = s-1/2

In the first equality we put in s-1/2 instead of a and get 2(s-1/2) = s

2s-1 = s

s = 1

Note: this is an example of infinite geometric series...
S∞=a1/(1-r)
given:1/2, 1/4, 1/8
r=1/2
a1=1/2
substitute
S∞=(1/2)/(1-(1/2))
S∞=(1/2)/(1/2)
S∞=1

$\frac{1}{2}$ means half of 1.so, the first sequence covered half of 1.

and, $\frac{1}{4}$ means quarter of 1.so, the second sequence covered half of the half.

and, by this way ,they are going to cover the full 1.

Here's how I solved.

Consider the maclaurins' series expansion of

(1-x) ^ -1 = 1 + x + x^2 + x^3 + x^4 + ------------------------------------

put x = 1/2;