An Intuitive Derivation/Proof of the Sum of an Infinite Geometric Series based on Zeno’s Paradox

A geometric series is a series of numbers where each number in the series is equal to the previous number multiplied by a constant multiplication factor. For example: 2, 4, 6, 8, 16… is a geometric series with a constant multiplication factor of 2.

The sum to infinity of such a sequence, then, can be represented as:

sum.PNG

where a is the first number in the series and r is the constant multiplication factor.

In most cases, this number will not converge to a finite value, however when r is less than 1 it can be shown that the sum has a finite solution. It is well known that the sum of the series 1, 1/2, 1/4, 1/8, 1/16… (a=1, r=1/2) is equal to 2. This can be calculated using the well known equation:

sum2.PNG

Until recently, I did not have an intuitive proof of this equation until I was reading a book about the Greek philosopher, Zeno. The link between infinite geometric series and Zeno’s paradox is not at all revolutionary, but I have not before seen it used as a proof for the infinite geometric sum – so here is a (not rigorous) derivation of the infinite geometric sum using Zeno’s paradox.

Imagine a race between a tortoise and a man. Of course, the tortoise gets a 1 mile headstart, but the man runs twice as fast as the tortoise. Initially then, the man is 1 mile behind the tortoise. After the race has begun the man runs 1 mile to arrive at where the tortoise began – but in the meantime, the tortoise has run another 1/2 mile (because the tortoise is half as fast as the man). Now the man runs a further 1/2 mile to catch up with the tortoise, but in the meantime the tortoise has run a further 1/4 mile. Later still, the man runs another 1/4 mile and the tortoise runs 1/8 of a mile (Will the man ever catch the tortoise? Yes, of course but this is the basis of Zeno’s Paradox). As you can see, we are generating a geometric sequence: 1, 1/2, 1/4, 1/8… and all the while the gap between the man and the tortoise is growing ever smaller. On top of this, by changing the ratio of speeds between the man and the tortoise we could generate any geometric sequence we want. For example, if the man was 3 times faster than the tortoise we would generate the sequence 1, 1/3, 1/9, 1/27, 1/81…

In the general case,  the multiplication factor, r, is simply a ratio of the two speeds:

sum7

Where vt is the velocity of the tortoise and vm is the velocity of the man.

If we went on for ever, the man would get arbitrarily close to the tortoise until the distance is infinitely small. We also know intuitively that at some point the man catches the tortoise. Therefore the sum of the distances in the infinite sequence would have to be equal to the distance traveled before the man catches the tortoise. Therefore, using basic mechanics we can calculate at what distance exactly the man catches the tortoise and hence the sum of the sequence.

The distance traveled by the tortoise, xt, and by the man, xm, at time, t, can be described by the equations:

sum5

The time at which the man catches up with the tortoise can be found by setting xm=xt. Then using that time we can find the distance traveled when the man catches the tortoise, x:

sum4

Using our definition for the constant multiplication factor, r, above, this becomes:

sum6

As you can see, this matches up with the equation at the beginning of the article without the a factor which is 1. Introducing a different starting value, rather than 1, is trivial. The starting value is the same as the headstart that the tortoise gets, because this is the first distance that the man must run.

As you can see, we have therefore calculated the sum of an infinite geometric series simply by finding the point at which a man overtakes a tortoise in a race. Hopefully this has explained the derivation of the sum of an infinite geometric series in a way that is a bit more intuitive than is often taught.

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