Add new comment

I think, rearranging (1-1+1-1+1-1...) as (1+1+1+1...) - (1+1+1+1...) is not valid because of the "shifting" of values, and decomposing of one infinity into two. If you insist on "proving" equality to 0 - there is an easier way: just use parenthesis like this: (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... which is "clearly" zero. Right? Not really.

This has been bugging me all day, and the best "intuitive" explanation may be based in physics: If you draw (1-1+1-1+1-1...) on a graph assuming it's some physical value over time - it's easy to see that 1/2 is the center of oscillation. So, even though the graph never converges to 1/2 - in the infinity it may as well converge. Given the example with a light bulb.. which is ON or OFF, in the infinity, the bulb would be neither ON or OFF - it would be half-bright. If you start sequence with -1, you get an oscillating line around -1/2. So, that makes sense too. If you start doing tricks like (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... -- it's easy to see that the trick here is selectively collapsing time intervals to 0, which doesn't make sense in the physical sense.

I started today thinking that (1-1+1-1+1-1...) = 1/2 was a fallacy, but now I think it's actually true, and it starts to make sense. However, I'm still to make the leap to understanding how (1+2+3+4+5+...) = -1/12 can be a useful fact, even if it's mathematically correct.

Filtered HTML

  • Web page addresses and email addresses turn into links automatically.
  • Allowed HTML tags: <a href hreflang> <em> <strong> <cite> <code> <ul type> <ol start type> <li> <dl> <dt> <dd>
  • Lines and paragraphs break automatically.
  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.

  • What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.

  • Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!

  • How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?

  • Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.

  • PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.