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That doesn't really show that it's amazing. You could say the same about x-x=0, even for large x. Or about any converging series. Unless all of that is amazing

That doesn't really show that it's amazing. You could say the same about x-x=0, even for large x. Or about any converging series. Unless all of that is amazing too - I wouldn't know, not being a mathematician.

AlEinstein 7y ago
On the face of it, if someone told me a series converged on a small value like -0.4, I wouldn't expect to encounter 43 million along the way. It's easy to dismiss things as uninteresting but HN is a forum for nerds and I'd conjecture that things like this are interesting for that group.
- wruza 7y ago
  Honestly, that was either too trivial, or underexplained for it to be a nerd-content. Maybe I’m watching too much pbs and numberphile, but nothing seemingly amazing here.
  - mark-r 7y ago
    You're obviously just jaded. I agree with the post, I'm surprised when a large list of seemingly random large numbers cancels out like that. The secret of course is that they aren't random, even though they appear so to the naked eye.
boyobo 7y ago
Write down the first 50 terms of the series for e^{-x}. Go back in time and tell Pythagoras to plug in x=2,3,4,5,6,7,8,10,... Each of the terms is a rational number, and the each of the sums will be an extremely small rational number (numerator will be tiny compared to denominator). Don't you think Pythagoras would be impressed and mystified? Would he be able to come up with an explanation for this `coincidence'? Now try the same with with x-x. That's not going to impress Pythagoras.
ridiculous_fish 7y ago
You gave two scenarios which are worth addressing: "x-x=0". Here we have two terms (x and -x). Their magnitude grows at the same rate, so it is not surprising that their difference is fixed and the two-term "series" "converges" to zero. "Any converging series." Consider the exponential growth function: e^x = 1 + x + x^2/2 + x^3/6 ... + x^n/n!. This grows fast, but we know that the factorial is super-exponential so the terms must approach zero for any fixed x. If the series converges (true though not obvious) then the value must increase very quickly. Now consider the exponential decay function. Unlike x-x, the terms are raised to different powers, so they grow at different rates. And unlike the exponential growth function, the series converges to zero in the limit. The "miracle" is not the convergence but how it happens: terms growing in opposite directions at wildly different rates. It's as if a lopsided spaceship chaotically fired rockets at full power in all directions, and happened to stay exactly still.
- schoen 7y ago
  > and happened to stay exactly still Or, perhaps, happened to make a soft landing on a specific planet!
- lopmotr 7y ago
  I see, it's more amazing than my simple examples. But why is it more amazing as this presumably uninteresting series I just made up sum(n=1...infinity, 1/2^n - 1/(2^n+1)) That also has alternating terms that grow differently in opposite directions and have huge numbers, yet they cancel out so that it converges to something. Is that uninteresting because each term in a pair has the same exponent, or because it doesn't converge to something that's closer to zero as something in the terms increases? Maybe you can't easily just make up a converging series that has all the features they listed and that uniqueness makes it interesting?
  - boyobo 7y ago
    I would say it's not as interesting because a high-schooler (or Pythagoras) could easily verify the fact AND they would easily be able to explain how you came up with that sum. One of my metrics for determining if a fact F is interesting is if P(F) is much larger than V(F). Here P(F) is, vaguely speaking, the amount of effort required to "explain" or "generalize" the fact, and V(F) is the amount of effort required to verify the fact. In the case of e^{-x}, each individual example can be verified by an elementary schooler so V(F) is very small. On the other hand, I don't see how you could explain the phenomena of convergence to 0 without teaching the elementary schooler a large part of calculus, so P(F)/V(F) is large.

AlEinstein 7y ago

On the face of it, if someone told me a series converged on a small value like -0.4, I wouldn't expect to encounter 43 million along the way. It's easy to dismiss things as uninteresting but HN is a forum for nerds and I'd conjecture that things like this are interesting for that group.

wruza 7y ago
Honestly, that was either too trivial, or underexplained for it to be a nerd-content. Maybe I’m watching too much pbs and numberphile, but nothing seemingly amazing here.
- mark-r 7y ago
  You're obviously just jaded. I agree with the post, I'm surprised when a large list of seemingly random large numbers cancels out like that. The secret of course is that they aren't random, even though they appear so to the naked eye.

boyobo 7y ago

Write down the first 50 terms of the series for e^{-x}. Go back in time and tell Pythagoras to plug in x=2,3,4,5,6,7,8,10,... Each of the terms is a rational number, and the each of the sums will be an extremely small rational number (numerator will be tiny compared to denominator). Don't you think Pythagoras would be impressed and mystified? Would he be able to come up with an explanation for this `coincidence'? Now try the same with with x-x. That's not going to impress Pythagoras.

ridiculous_fish 7y ago

You gave two scenarios which are worth addressing: "x-x=0". Here we have two terms (x and -x). Their magnitude grows at the same rate, so it is not surprising that their difference is fixed and the two-term "series" "converges" to zero. "Any converging series." Consider the exponential growth function: e^x = 1 + x + x^2/2 + x^3/6 ... + x^n/n!. This grows fast, but we know that the factorial is super-exponential so the terms must approach zero for any fixed x. If the series converges (true though not obvious) then the value must increase very quickly. Now consider the exponential decay function. Unlike x-x, the terms are raised to different powers, so they grow at different rates. And unlike the exponential growth function, the series converges to zero in the limit. The "miracle" is not the convergence but how it happens: terms growing in opposite directions at wildly different rates. It's as if a lopsided spaceship chaotically fired rockets at full power in all directions, and happened to stay exactly still.

schoen 7y ago
> and happened to stay exactly still Or, perhaps, happened to make a soft landing on a specific planet!
lopmotr 7y ago
I see, it's more amazing than my simple examples. But why is it more amazing as this presumably uninteresting series I just made up sum(n=1...infinity, 1/2^n - 1/(2^n+1)) That also has alternating terms that grow differently in opposite directions and have huge numbers, yet they cancel out so that it converges to something. Is that uninteresting because each term in a pair has the same exponent, or because it doesn't converge to something that's closer to zero as something in the terms increases? Maybe you can't easily just make up a converging series that has all the features they listed and that uniqueness makes it interesting?
- boyobo 7y ago
  I would say it's not as interesting because a high-schooler (or Pythagoras) could easily verify the fact AND they would easily be able to explain how you came up with that sum. One of my metrics for determining if a fact F is interesting is if P(F) is much larger than V(F). Here P(F) is, vaguely speaking, the amount of effort required to "explain" or "generalize" the fact, and V(F) is the amount of effort required to verify the fact. In the case of e^{-x}, each individual example can be verified by an elementary schooler so V(F) is very small. On the other hand, I don't see how you could explain the phenomena of convergence to 0 without teaching the elementary schooler a large part of calculus, so P(F)/V(F) is large.