An intuitive reasoning for 1+2+3+4+5… + ∞ = -1/12? [duplicate]

I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww

In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this a joke? Or does it contain somewhere within it a flaw?

Now, I know that this channel and the people running it are by profession researchers so it does make me think whether this is true.

but at the same time, you are adding positive numbers on one hand and getting a negative sum?

The video also shows how 1-1+1-1+1-1.... =1/2 and that is counter intuitive too, how can you get a fraction when you are only adding together integers? I do however, painfully accept this as the truth because this is an infinite G.P. with r= -1. and I can get the answer using the formula.

1. Is this actually true or altogether false, or a limitation to the current system of mathematics and accepted as true only because otherwise, mathematics would be proven wrong?

2. Is there any intuitive explanation to this (both the series that are mentioned), or can I only see it using mathematics?

3. The video mentions that their are actual uses in physics for this. What are they?

I have read other answers and the opinions in them are conflicting, some say it is false and some say that it is true. Also, other questions don't seem to answer all hree of the questions posed.

-

marked as duplicate by DonAntonio, gt6989b, Harald Hanche-Olsen, Nicholas R. Peterson, Peter KošinárJan 9 at 18:37

The idea underlying these counterintuitive "sums" isn't the--flawed--notion that the limit as $n$ goes to infinity of the sum of positive integers from $1$ to $n$ is, in fact, $-1/12$, but rather that there's a consistent way to assign a real number to certain formal infinite series that allows us to do computations with them even when sums for those series don't exist. –  Nick Jan 9 at 17:39
There are several ways to "define" the value of a divergent series $\sum_{n \geq 1} a_n$ using complex analysis. One way is to study the power series $f(z) = \sum_{n \geq 1} a_nz^n$ and the other is to study the Dirichlet series $g(s) = \sum_{n \geq 1} a_n/n^s$. Formally, $\sum a_n$ equals $f(1)$ and $g(0)$. Analytically, if $f(z)$ converges for small $z$ and has an analytic continuation to $z = 1$, you could define $f(1)$ to be that value. And if $g(s)$ converges for ${\rm Re}(s)$ large and has an analytic continuation back to $s = 0$, you could define $g(0)$ to be that value. That's all. –  KCd Jan 9 at 19:04