This wonderfully entertaining video giving a proof for why the sum of the natural numbers is -1/12 has been viewed over 1.5 million times. It just shows that there is a hunger for interesting and well explained math and science content out there. Now, we all know that the sum of all the natural numbers is infinite but the beauty (insidiousness) of infinite numbers is that they can be assigned to virtually anything. The proof for this particular assignment considers the subtraction of the divergent oscillating sum from the divergent sum of the natural numbers to obtain . Then by similar trickery it assigns . Solving for gives you the result . Hence, what you are essentially doing is dividing infinity by infinity and that as any school child should know, can be anything you want. The most astounding thing to me about the video was learning that this assignment was used in string theory, which makes me wonder if the calculations would differ if I chose a different assignment.
Addendum: Terence Tao has a nice blog post on evaluating such sums. In a “smoothed” version of the sum, it can be thought of as the “constant” in front of an asymptotic divergent term. This constant is equivalent to the analytic continuation of the Riemann zeta function. Anyway, the -1/12 seems to be a natural way to assign a value to the divergent sum of the natural numbers.