summation - How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6ldots$” to infinity = “$-1/12$”?

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?

I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (http://youtu.be/uSwJuOPG4FI). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:

Let’s go to some of the things Dr. Craig talked about. In fact, the
existence of infinity, which he talked about which is
self-contradictory, is not self-contradictory at all. Mathematicians
know precisely how to deal with infinity; so do physicists. We rely on
infinities. In fact, there’s a field of mathematics called “Complex
Variables” which is the basis of much of modern physics, from
electro-magnetism to quantum mechanics and beyond, where in fact we

learn to deal with infinity; without the infinities we couldn’t do the
physics. We know how to sum infinite series because we can do complex
analysis. Mathematicians have taught us how. It’s strange and very
unappetizing, and in fact you can sum things that look ridiculous. For
example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,
what’s the answer? “-1/12.” You don’t like it? Too bad! The
mathematics is consistent if we assign that. The world is the way it
is whether we like it or not.

-- Lawrence Krauss, debating William Lane Craig, March 30, 2011

Source: http://www.reasonablefaith.org/the-craig-krauss-debate-at-north-carolina-state-university

CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post: https://physics.stackexchange.com/questions/92739/how-does-the-sum-of-the-series-1-2-3-4-5-6-to-infinity-1-12

EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.