Thread: Math trivia
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Old 03-22-2011, 06:20 PM
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ceptimus ceptimus is offline
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Default Re: Math trivia

The harmonic series (and modifications thereof):

It's interesting that sometimes, we can add an infinite number of numbers together and get a finite result. Perhaps the most obvious example is where each term in the series is half as big as the one before:

1/2 + 1/4 + 1/8 + 1/16 + ...

This is the series made famous by Zeno, and although it can seem confusing at first, it's pretty obvious that, in this case, the sum of all the terms is one. Each extra term added takes us 'half the remaining distance' towards one, but it's clear that even after an infinite number of terms we can never get past one.

Now lets look at the harmonic series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

Each term is smaller than the one before, so it looks like it might approach some total as we add more and more terms, but never pass that total.

So we begin adding terms and find that after the first 10 terms we reach 2.929, after 100 terms we've got to 5.187 and after 200 terms we have 5.878 - perhaps we're heading towards 2 x PI which would be 6.283?

But no, after 300 terms that value is passed and the total keeps creeping up - albeit very slowly - after the first thousand terms the total is still less than 7.5

The amazing thing is that although the rate of increase of the total quickly slows to a snail-like crawl, it's a snail that never gives up, so the total will eventually pass 10, 100, a billion, any number you can name - the total is, in fact, infinite.

Now let's modify the series slightly - pick any number you like, say 314159 in honour of PI. Cross out all the numbers in the harmonic series that contain that number. So the first term we cross out is at the 314160th place and the next is one million places after that: 1/1314159 then another million places: 1/2314159 and so on...

The total was infinite before, so you might think that after crossing out these numbers the total would still be infinite. But no! Schmelzer and Baillie discovered an efficient way of working out such results, and found that the total with all the '314159 numbers' missing is 2302582.33386378260789202376 (rounded to the last decimal place!)

In general, you can choose any rule you like for crossing out numbers, and it will be enough to topple the harmonic series from having an infinite sum to a finite one (mathematicians would say it changes from being a 'divergent series' to a 'convergent' one).

One reason why this is so, is that once we get to really huge numbers - ones with a few billion digits - then there is a very good chance that they will contain, say, '314159' somewhere - so we end up crossing out most of the numbers as we get further into the list.

But even if we make our chosen string of digits much longer - say we only cross out those numbers where the digit '4' occurs a billion times in a row - then that is still enough to change the divergent harmonic series into a convergent one!

Last edited by ceptimus; 03-22-2011 at 08:27 PM.
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