In this video of numberphile they show a series of calculations to show that the sum of natural numbers is -1/12. Then I tried to proof other things using these mathematics techniques and I found the following... is it correct? there are a specific rules when trating these numbers?
The sum of natural numbers is:
$S_1 = 1+2+3+4+5+\cdots = -\frac{1}{12}$
Another summatory:
$S_2 = 1-1+1-1+\cdots = \frac12$ (proof)
And I want to calculate $S_3$:
$S_3 = 1+1+1+1+\cdots $
Lets add $S_1$ and $S_2$:
\begin{align} S_1=&1+2+3+4+5+6+7+8+9+10+\cdots\\ S_2=&0+1+0-1+0+1+0-1+0+1+\cdots\\ S_1+S_2 =& 1+3+3+3+5+7+7+7+9+11+\cdots\\ S_1+S_2 =& 1+5+9+13+\cdots\\ &3(3+7+11+15+\cdots) \end{align}
Adding $2S_3$ to this sum: \begin{align} S_1+S_2+2S_3 =& 1+5+9+13+\cdots+2+2+2+2+\cdots\\ &3(3+7+11+15+\cdots)\\ S_1+S_2+2S_3 =& 3+7+11+15+\cdots\\ &3(3+7+11+15+\cdots)\\ S_1+S_2+2S_3 =& 4(3+7+11+15+\cdots)\qquad (Eq.1) \end{align}
On the other hand: \begin{align} S_1 &= 1+2+3+4+5+\cdots\\ S_1 &= 0+1+2+3+4+\cdots\\ 2S_1 &= 1+3+5+7+9+\cdots \end{align}
And also
\begin{align} 2S_1 &= 2(1+2+3+4+5+\cdots)\\ 2S_1 &= 2+4+6+8+10+\cdots \end{align}
Then if we add these results
\begin{align} 2S_1 &= 1+3+5+7+9+\cdots\\ 2S_1 &= 2+4+6+8+10+\cdots\\ 4S_1 &= 3+7+11+15+19+\cdots \end{align}
Replacing in Eq.1 we have
\begin{align} S_1+S_2+2S_3 &= 4(4S_1)\\ S_3 &= \frac{15S_1-S_2}2\\ S_3 &= \left(-15\left(\frac1{12}\right)-\frac12\right)\frac12\\ S_3 &= -\frac{21}{24}\\ 1+1+1+1+\cdots &= -\frac{21}{24} \end{align}
However this sum corresponds with Riemann zeta function for $s=0$, which is $\zeta(0)=-\frac12$.
My question is, there is something wrong? there are specific rules when trating with these summatories?