TITLE: Is $\sum_{n=1}^{\infty} 1 = -\frac{3}{12}$ true?
QUESTION [16 upvotes]: Here is how I derived this...
$$1+2+3+4+...=-\frac{1}{12}$$
$$2+4+6+8+...=2(1+2+3+4+...)=2(-\frac{1}{12})=-\frac{1}{6}$$
Thus $1+3+5+7+...=-\frac{1}{6}-(1+1+1+...) $ because $S_{odd}=S_{even}-(1+1+1+...)$ 
Also $S_{odd}=(1+2+3+4+...)-(2+4+6+8+...)=-\frac{1}{12}+\frac{1}{6}=\frac{1}{12}$
Thus $\frac{1}{12}=-\frac{1}{6}-(1+1+1+...)$
$$1+1+1+...=-\frac{3}{12}$$
What I think my mistake was (if I have one) is where I assume the sum of all numbers is half the sum of all even numbers; although, it should work since there is nothing two times infinity. 
Please leave  simple solution (im only 15 years old) to why this is false: Wikipedia says that $\sum_{n=1}^{\infty} 1 = \infty$.
Edit: I know now that the above notion is false. I recently watched a Numberphile video proving $\sum_{n=1}^{\infty} n = -\frac{1}{12}$. I followed the same line of reasoning to derive the untrue $\sum_{n=1}^{\infty} 1 = -\frac{3}{12}$ I'll admit; I was ignorant in believing so and I apologize for wasting your time.
Thanks for the answer kindly explaining why I was wrong. I guess I should watch this video to un-brainwash me. Sorry again.

REPLY [2 votes]: No, it isn't true, at least not with the standard definition of infinite summation.
The sum
$$\sum_{n=1}^{\infty} 1$$
is either undefined, or equals $\infty$. "$\infty$" is an element in the "extended real number system", not a real number, and has the property that
$$\infty - \infty$$
is undefined. Both your $S_\mathrm{odd}$ and $S_\mathrm{even}$ also sum to $\infty$, hence your subtraction of them invokes the above form and is undefined. Conversely, you can consider this as a proof that such a subtraction has to be undefined, else it will lead to inconsistencies in the number system - in particular, your last equality is
$$\infty = -\frac{3}{12}$$
which is nonsense of the same form as
$$2 = 1$$
and your "proof" is analogous to the the phenomenon which occurs in "fake proofs" of the above involving the similarly-undefined division by $0$.