Yes Virginia, you can divide by zero. (in some circumstances)
This is from a textbook on complex analysis. CLICK HERE
Or try entering 1/0 on Wofram Alpha CLICK HERE
focal settings on a camera lens include infinity. Oh my.
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
Albert Einstein (1879 - 1955)
The sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this:
First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and thus, X = 1/2.
Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and thus, Y = X/2 = 1/4.
Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and thus, Z = -Y/3 = -1/12.
In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't.
All this may seem rather contrived and somewhat like a Jedi Mind Trick.
So where can you find a solid respectable way to show this result?
Wolfram Alpha is a very powerful online computation engine.
The infinite series 1+2+3+4…. can in fact be represented using the Rieman Zeta Function.
Typing in zeta(-1) creates the infinite series 1+2+3…..and returns the answer -1/12.
The mathemetician Srinivasa Ramanujan was also aware of this. CLICK HERE.
A derivation of this appears in one of his notebooks