TITLE: Why can infinite series be summed different ways to get different results?
QUESTION [5 upvotes]: $$S = 1 - \frac12 + \frac13 - \frac14 + \frac15 - \frac16 + \frac17 - \frac18 + \frac19 - \frac1{10} + \frac1{11} - \frac1{12}\ldots\text{(to infinity)}$$
Rearranged, this series looks like:
$$S = \left(1 - \frac12\right) - \left(\frac14\right) + \left(\frac13 - \frac16\right) - \left(\frac18\right) + \left(\frac15 - \frac1{10}\right) - \left(\frac1{12}\right) + \left(\frac17 - \frac1{14}\right) \ldots\text{(to infinity)}\\
 S = \left(\frac12\right) - \left(\frac14\right) + \left(\frac16\right) - \left(\frac18\right) + \left(\frac1{10}\right) - \left(\frac1{12}\right) + \left(\frac1{14}\right) \ldots\text{(to infinity)}$$
This rearranged infinite series contains every number that the original infinite series had.
Further,
$$ 2S = 1 - \frac12 + \frac13 - \frac14 + \frac15 - \frac16 + \frac17 - \frac18 + \frac19 - \frac1{10} + \frac1{11} - \frac1{12} \ldots\text{(to infinity)}$$
Thus: $2S = S$
$2 = 1$
Mathematics disproven.  Sorry.
Jokes aside, I know that infinite series can be calculated in different ways to get different results.  My question is: Why?  While it makes sense with Grandi's and similar series, it doesn't make sense to me for a series whose final term is $\frac1\infty = 0$.

REPLY [8 votes]: This is a good opportunity to point out something that many people don't get, or at least get accustomed to glossing over. When you write something like $S = 1 - 1/2 + 1/3 - 1/4 + 1/5$, you really are saying that $S$ is a sum; you can explicitly add the terms (and get $47/60$ in this specific case).
On the other hand when you write something like $S = 1 - 1/2 + 1/3 - 1/4 + 1/5 - \cdots$, this is a completely different animal. You can't explicitly add the terms because there are infinitely many of them. $S$ is not a sum, even though it is traditional to say so (I call it a sum as well). It may be spoken of as an "infinite sum", and it's convenient to regard it that way be cause in the right conditions it acts like a sum, but it is not--it is actually a limit.
For each $n$, the partial sum $S_n = 1 - 1/2 + \cdots + 1/n$ is indeed a sum which can be computed by explicitly adding the terms. The "infinite sum" $S$ is not a sum at all, but rather the limit $\lim\limits_{n\rightarrow\infty}S_n$, if it exists.
Often it is fine to forget about that and pretend $S$ is really a sum, but that can get you in trouble sometimes. So always keep in the back of your mind the notion that infinite sums are really limits of finite sums.