Contradiction in calculating the sum of Grandi's series

Question

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.

Answer 1

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.

Answer 2

This series diverges. It's not possible to assign it a meaningful answer via standard series techniques. There are generalized notions of summation such as Abel, Cesaro and Ramanujan which can assign meaningful values to some otherwise divergent series (in the usual sense). In this case, Cesaro summation gives a value of $\frac{1}{2}$ for this series like you got initially albeit by different reasoning. See this Wiki article.