I don't know much about infinite series (actually I don't know anything about it), but I just happened to think of this.

So I was reading Wikipedia and I learned about Grandi's series. It is a series of -1 and +1:

$$ S = 1-1+1-1... $$

To find out its sum, one can do:

$$
\begin{align}
1-S &= 1-(1-1+1-1...) \\
1-S &=S \\
S &= \frac{1}{2}
\end{align}
$$

However, by using the same logic, one can also say:

$$
\require{cancel}
\begin{align}
1-S &= 1-(1-1+1-1...) \\
S &= 1-S \\
&= 1-1-S \\
&= 1-1-1-S \\
&= \cancel{-\infty} \text{A very big negative number} - S \\
\text{let N be the very big negative number:} \\
-N-S &= S \\
-N &= 2S \\
S &= -N
\end{align}
$$
So the sum of the series tends to be a infinitely small number according to its logic. So what is going on here and is this a contradiction? I thought it is $\frac{1}{2}$ but now it seems like $-\infty$ is also a legit answer.