Is this a legitimate geometric proof that the infinite sum of 1/(2^n) = 1?

[Steven G]

I meant that The area of a 1 by 1 square is 1(SQUARE unit)

 

[Steven G]

Okay, thanks a lot!

You should fill in the words! First you shade in half the area. Then you shade in half of the remaining 1/2 half, ie you shade in 1/4 of the area leaving 1/4 of the area still unshaded. Of the 1/4 of the remaining area you shade in 1/2 of that area leaving the other half (1/8 of the area) unshaded. ....

 

[blamocur]

Yes, but I want to learn different kinds of proofs than using the definition.

I'd suggest getting a good command of "classical" proofs before searching for more exotic ones. In the case of this series you can try a different approach: 1) prove that the sum is limited from above; then the fact that it is monotonic means it must converge; then apply post #14. But "geometric proofs" rarely qualify as formal ones.

 

[Mates]

You should fill in the words! First you shade in half the area. Then you shade in half of the remaining 1/2 half, ie you shade in 1/4 of the area leaving 1/4 of the area still unshaded. Of the 1/4 of the remaining area you shade in 1/2 of that area leaving the other half (1/8 of the area) unshaded. ....

I will take a stab at it.

Every area that is not being filled by a blue area gets covered by an orange area. Therefore, the total area of 2 has to be completely filled.