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Infinite mathematical sequences are like snakes. Some are clearly very scary and you should stay well away from them. Some are harmless enough. Some look harmless, but can turn round and bite you painfully.
Here's a harmless snake to get us started. There was a Greek philosopher called Zeno. Like an alarming number of philosophers, he was convinced of the truth of something that is clearly false, and spent his life trying to prove it. His particular belief was that motion is impossible. To prove it, he pointed out that to get from A to B, you (or a frog - it makes no difference) have to get halfway, then another quarter of the way, then another eighth, halving your distance from B at every stage. This involves an infinite number of steps, so you will never get there.
However, you can view the steps as an infinite sequence: 1/2, 1/4, 1/8, 1/16, and so on. Even though there is an infinite number of terms in the sequence, if you add them together, which you can do mathematically though not on a computer, they converge to exactly one.
Clearly, this won't happen with a sequence that doesn't keep getting smaller. 1+1+1+1+... is obviously infinite. An unambiguously harmful snake.
Still, it appears that if the terms in an infinite sequence keep getting closer to zero, and will eventually get as close as you like, you can add them together and arrive at a finite total. Right? Wrong.
Here is our first innocent-looking but dangerous snake: the reciprocals, 1/1, 1/2, 1/3, 1/4 and so on. Try adding those together and you will see the total gets bigger and bigger. This is called the harmonic series, and it's anything but harmonious.
Let's vary things a bit: instead of calculating 1/1+1/2+1/3+1/4+..., let's change every other plus sign to a minus. So, 1/1-1/2+1/3-1/4+1/5-1/6... Does that help? Yes, now the snake is harmless again. The total is...wait for it...ln(2), where "ln" means "log to the base e". e is one of those weird transcendental numbers that becomes ever more common once you get past GCSE. And here it pops up in the sum of a decidedly non-transcendental series. Very strange.
What about the sum of the reciprocals of all the prime numbers, 1/2+1/3+1/5+1/7+1/11+...? We know the primes get more and more sparse as they go on, so surely this series should shrink fast enough to give us a nice finite sum? Nope, it's infinite.
How about squaring the numbers in the harmonic series? 1/1+1/4+1/9+1/16+...? Now we're in luck again. And the total is another unexpected one: nothing to do with e or logs this time, it's π²/6. Would you have guessed it was anything like that? I certainly wouldn't.
Perhaps the strangest such sequence is what you get by adding the reciprocals of the Fibonacci numbers. The Fibonacci sequence starts with two ones, and then every subsequent value is the sum of the previous two. So: 1, 1, 2, 3, 5, 8, 13 and so on. Take the reciprocals and add them together: 1/1+1/1+1/2+1/3+1/5+1/8+1/13+... The sum is finite, and the value is about 3.3599. It has a special name, ψ, pronounced "psi". It looks transcendental, and certainly irrational (not expressible as a fraction) but even Erdős, the mathematician famous for his 1500 papers and 500 coauthors, was unable to prove it; the world of maths had to wait until 1989 for that particular milestone.
Even simple maths can sometimes give you surprising results. This suggests there are hidden connections waiting to be discovered, which is a large part of why it's fun. If you're brave enough, here is the entrance to the rabbit hole.
If you are interested in discussing some sessions to learn more about snakes mathematical sequences and other topics, please get in touch.
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