1

I am 13 years old and this question came to my mind during recess.

And I think I have an answer but I would like your approval.

-The question:

If I have a 10-meter-high ceiling and a 1cm thick sheet of paper; if you have the possibility (in an ideal case) to place infinite slips of paper on top of each other, with the rule that each slip of paper must be half of the thickness of the previous one.

Will the leaflets ever reach the ceiling?

Practically: if we had the ability to place infinite slips of paper on top of each other, in an ideal world, it would look something like this:

(This is the thickness of the first 9 leaflets but it goes on forever)

$$1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + 0.015625 + 0.0078125 + 0.00390625 + \dots$$

Which if taken to infinity might seem possible, but let's think a moment longer small...

Instead of having to get to the ceiling let's say the thickness of all the leaflets has to reach 2cm (twice the initial thickness of the first leaflet) will the leaflets ever get there?

The answer is Yes! , but in an ideal world.

The leaflets in this case will have the thickness that will be identical to that of the first leaflet, which in this case is 1cm.

-Explanation:

If we try to add up in fraction form the numbers previously written, without counting the first leaflet put, we find that the number that comes out in fraction form is always closer to the first leaflet:

$$1/1; 1/2; 1/4; 1/8; 1/16; 1/32; 1/64; 1/128; 1/256; 1/512; 1/1024; 1/2048; 1/4096 \dots$$

$$2048 + 1024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 / 4096 = 4095 / 4096 \approx 0,999755$$

If we continue, the result will be closer and closer to the integer: $4095.875/4096 \approx 0.999969$ and if we repeat this calculation to infinity it will become $0.99$ periodic or in mathematics 1 which is the thickness of the first leaflet. So we can conclude that in an ideal world if we could put infinite leaflets fi thickness half than the previous one the thickness would be the same as the thickness of the first leaflet.

Thanks for having read all o this, now i'm really curios about a respond.

Tina
  • 1,170
Kevin Pedone
  • 21

1 Answers1

0

The situation you are describing can be modeled by this summation for the first $n$ sheets of paper:

$\displaystyle\sum_{k=0}^{n} \frac{1}{2^k}$

When we add an infinite amount of terms (sheets of paper), we get this series: $\displaystyle\sum_{k=0}^{\infty} \frac{1}{2^k}$, which is convergent (i.e. it equals a specific value) and converges to 2, meaning that the stack of paper will never go any higher than 2cm even with an infinite amount of sheets.

Hope this helps!

COIsCurious
  • 155