Maths problem to solve

[Aractus]

Okay, so I saw Numberphile use this little example as a throwaway example in one of their videos, so I thought I'd re-frame it as a problem and see if anyone here can solve it.

Imagine you have a pet ant, and he's your favourite pet. You absolutely love him, and you want to keep him entertained while you're away. In the same way that hamsters love running on wheels, your ant loves to walk around the circumference of elastic bands. He thinks it's terrific, but you have to keep buying more elastic bands for him because he never wants to walk the same elastic band twice. At the moment you use elastic bands that are stretched out to a circumference of one metre, and your ant walks around reach one in 100 seconds - it takes him 1 second to move 1cm. You currently need to supply your athletic ant with enough entertainment to last him nine hours a day, which means he goes through 324 huge 1m elastic bands per day!

We'll assume that your ant has been blessed with immortality, as well as infinite stamina. A cleaver entrepreneur has just invented a new type of elastic band that can keep on stretching out forever. You come up with an ingenious idea - if you stretch it out by 1 metre per second it should last your ant a very long time, and you won't have to keep buying hundreds of brand new 1m circumference elastic bands for him every day. In fact you can just buy one and by the time your pet ant finishes, then you can just get another.

So you get the new elastic band that can be stretched forever, it starts with a circumference of 1 metre, and you place your pet ant on it, and he starts walking around the circumference. Every time your ant walks 1cm you will stretch out the elastic band by another metre. So it starts with a circumference of 1 metre, the ant walks 1cm, you stretch it out by a metre, it's now two metres in circumference and the ant walks another centimetre, and then it gets stretched out to three meres, and he walks another centimetre, and this process will keep on going until he completes his journey around the elastic band, just like this:

https://blog.aractus.com/videos/numberphile-ant.mp4

Video © Numberphile

I was going to take a screenshot and show you that, but then I thought I'll just show you a short section of the numberphile video so you can see the elastic band stretch after each 1cm the ant moves.

So the question is this: can your ant ever complete his circuit around the circumference of the elastic band? If not, why not, and if he can how long will it take and how big will the elastic band be at the end?

Assume of course that he never stops walking, 24 hours a day, 7 days a week, until he completes, if he completes his gigantian journey.

(edited to add...) And FYI, if you think you'll just find the Numberphile video and use their answer, the answer they give to this problem is wrong, and I can show you the calculation that proves that. In fact not only is their answer wrong, it's spectacularly wrong. Their answer is not even close to correct. But then again half the stuff they show you like "1 + 2 + 3 + 4 + ... = -1/12" is wrong, but in this instance it's clear they made a miscalculation in their answer to this "throw-away question". If you want a hint as to how to solve this problem, first construct a finite series to describe the problem using maths, and then solve for the value of that series.

[Cato]

You're going to have to explain why a finite series is necessary to solve a problem with no finite boundaries. You addressed the ant, but the. Umber of circumference expansions and the space it is expanding into aren't constrained. I suspect the answer involves some bastardization of the concept of infinity, particularly how it's used mathematically.

This is like the inverse of Zeno's Paradox. Something's amiss. I look forward to the answer. It had better not be that the elastic material is finite and that we'll eventually reach a point that the surface is so thin that the ant falls off.

[Aractus]

Well you're using part of an infinite series, but it's finite. Let's say we use the series of all the natural numbers up to the nth number:

N = 1 + 2 + 3 + 4 + ...

Right so if we use it as a finite series we can solve for the nth number quite easily. Let's say solve for n=10 - that's just:

N = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66

Assuming that we start from n=0. There's no funny business there. Once you've figure out the series you need you can look up how its value is calculated for n terms, and then calculate it.

If we want to know the size our elastic band has stretched to after 11 years, we would use this series:

N = 1 + 1 + 1 + 1 + ...

Basically just the number of seconds that has passed plus 1 (since it started with size 1m).

Approximately 347,124,097 meters. The question though is where is the ant, has he completed his journey, will he complete his journey and how long will it take him?

[Tiberius]

1 + 2 + 3 + ... = -1/12 is correct as a Ramanujam summation:

https://en.m.wikipedia.org/wiki/1_%2B_2_...%2B_4_%2B_⋯

[Aractus]

1 + 2 + 3 + ... = -1/12 is correct as a Ramanujam summation:

https://en.m.wikipedia.org/wiki/1_%2B_2_...%2B_4_%2B_⋯

It's not correct because it's not a sum it's a series. You can calculate the value of an infinite convergent series, but not a divergent one. The term as written is a bastardisation of the equals sign in which it means "is associated with" and not "is equal to". Anyway that's besides the point because for this problem you do not need to use some shifty value for an infinite divergent series, you just need to know the series and how to calculate the value to the nth term of the series.

[Tiberius]

Intuitively it seems that if an ant only moves at 1cm/s and you are stretching the rubber band by 1m/s, the ant will never be able to reach the other side.

However this is math, so I'm sure the answer is actually the opposite, and when you approach infinity the ant reaches the other side.

[Whateverist]

I must not be understanding the question. After one second the ant travels 1 cm but the distance left for him to travel increases from 100 cm to 199 cm. After two seconds, he will have traveled 2 cm but now has 298 cm left to go. How could the ant ever finish if there are no restraints on the capacity of the band to keep on stretching. Since the length of band left for the ant to travel will always be greater than before, how can we talk of him ever completing the band?

Since no mention is made of the space available for the band to stretch I have no idea if that is a consideration, but I assume not. What am I missing?

[Tiberius]

1 + 2 + 3 + ... = -1/12 is correct as a Ramanujam summation:

https://en.m.wikipedia.org/wiki/1_%2B_2_...%2B_4_%2B_⋯

It's not correct because it's not a sum it's a series. You can calculate the value of an infinite convergent series, but not a divergent one. The term as written is a bastardisation of the equals sign in which it means "is associated with" and not "is equal to". Anyway that's besides the point because for this problem you do not need to use some shifty value for an infinite divergent series, you just need to know the series and how to calculate the value to the nth term of the series.

You can calculate the sum of a divergent series using Ramanujam summation. That's what it was invented for. Its not a true sum of the series of course, but it works and is used to study actual divergent series'.

[Cato]

This is asinine. If for every step you take, 99 more steps are added to the journey you will never complete the journey. If you present math that suggests otherwise, your math is wrong. Inescapable.

[Aractus]

You can calculate the sum of a divergent series using Ramanujam summation. That's what it was invented for. Its not a true sum of the series of course, but it works and is used to study actual divergent series'.

Well the Numberphile video is wrong because they say "is equal to" not "is associated with".

I must not be understanding the question.  After one second the ant travels 1 cm but the distance left for him to travel increases from 100 cm to 199 cm.  After two seconds, he will have traveled 2 cm but now has 298 cm left to go.  How could the ant ever finish if there are no restraints on the capacity of the band to keep on stretching.  Since the length of band left for the ant to travel will always be greater than before, how can we talk of him ever completing the band?

Since no mention is made of the space available for the band to stretch I have no idea if that is a consideration, but I assume not.  What am I missing?

You're not missing anything, but you have incorrectly calculated the distance. After 1 second the ant travels 1cm, leaving him 99cm to go, and you stretch the elastic band to 2m. The distance in front of the ant is now 198cm (double), and the distance behind the ant is now 2cm. The he walks another cm, leaving him 197cm to go, and you stretch the elastic band to 3m. The distance in front of him is now 295.5cm, and the distance behind him now 4.5cm. So you can see the distance left to go is not quite as far as you thought (298cm).

Again please see my hint - construct a series that describes the distance the ant travels after n seconds accurately.