Contradiction in calculating the sum of Grandi's series 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.

If we know ahead of time that the series is (for example) Abel summable to $S$, then your manipulation is a perfectly valid demonstration that $S=1-S$, because the Abel sum respects such manipulations. That's not the problem.
The problem is relatively mundane: it occurs when you equate "$1-1-1-S=-N-S$". That's implicitly bracketing the subtraction as $((1-1)-1)-S$. But the derivation actually gives you the correct result as $S=1-(1-(1-S))$, which doesn't involve a big negative number.
