Does 1+2+3… really equal -1/12?

A recent Numberphile video claims that the sum of all the positive integers is -1/12. Bothered by that, Evelyn Lamb talks about what it means to assign a value to an infinite series and explains different ways of doing this.

http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/