Tuesday, September 6, 2011

Why 1+2+3+4+.....= -1/12

Speaker: Masoud Khalkhali, UWO
Time: Tuesday, September 27, 4:30-5:30 PM
Place: MC 107
As usual pizza and pops will be served after the talk in grad club.

Let us kickoff this year's Mathematics Pizza Seminar Series with some interesting piece of analysis. The title of my talk sounds like an utterly wrong statement! After all, the infinite series 1+2+3+4+.....is divergent and in fact diverges to infinity. So shouldn't we just write 1+2+3+4+....= infinite ? Of course we can. But then with the same, limited and narrow minded, understanding of summation we shall assign the same value, infinite, to a host of other very different types of series like 1+1+1+1+......or 1 +4 +9+16+25+....... etc. The point of my talk is that in doing so we are throwing away a wealth of information hidden in such divergent series. Information that can have practical implications for mathematics and its applications. This situation is a bit similar to set theory and cardinal numbers. Mathematicians used to think that there are only two types of numbers: finite and infinite. Of course, after Cantor, we know that there is a vast hierarchy of infinities and knowing about these different types of infinities is often very useful, though sometimes it creates its own problems. This is an important analogy that we should keep in mind.

One of the first people who realized the importance of divergent series and developed some techniques to sum such divergent series was Leonhard Euler. In fact Euler was of the opinion that any series is summable and one should just find the right method of summing it! In the last 250 years many summation techniques have been designed and there is vast theory of summability: Abel summation, Cezaro summation, Borel summation, zeta summation, etc.

This talk is a quick introduction to one such theory of summability: zeta function regularization. I will show how to compute infinite sums like the one on the title, as well as many others like
1+1+1+1+.....= -1/2

I will also discuss how infinite products like (all positive integers) can be defined and evaluated in some cases, e.g. =-1/2 log (2 \pi)

One of Euler's long standing goals in this area of math was to find (nowadays we say to define!) the alternating sum of factorials
1!-2! +3! -4! +5! -.....
I shall explain this and end up with Euler's surprising answer!


  1. Hey, I stumbled on this trying to search for the zeta function and some answers to how, 1+2+3... = -1/12. Anyway, you state we can say that summation is infinity. I'm ok with that. How in the heck can we say with a straight face it equals -1/12?? That answer gives us a negative number as a sum of positive number and also a fractional number less than the first number to be summed. It must be absolutely false! So why is it true?

  2. As 1-2+3-4+5-6+...=1/4(Hu,2). The sum of 1+3+5+...=1/12 and 2+4+6+...=-1/6, we must have (1+3+5+...)+(2+4+6+...)=1/12-1/6=-1/12. Proved

  3. As 1-2+3-4+...=1/4(Hu,2). The sum of 1+3+5+...=1/12 and 2+4+6+...=-1/6. Then (1+3+5+...)+(2+4+6+...)=1/12-1/6=-1/12.Proved