Infinite sums are not commutative.
This is perhaps one of the most surprising results about series,
and it is at the heart of the difference
between absolutely convergent and conditionally convergent series.
That is the topic of this video.
I'm going to start with an example.
Let's look at this series:
the alternating harmonic series 1 - 1/2 + 1/3 - 1/4, and so on...
This series is convergent, thanks to the alternating series test.
Now I'm going to compare with this other one, which is a reordering.
I am adding the very same terms, but I change the order.
Now I am adding two positive terms, then one negative,
then two positive, then one negative, and so on...
But the terms I am adding are the same.
If these were finite sums, they would be equal.
But they're infinite sums!
I'm not going to prove it,
but it is possible to show that the first sum adds to ln(2),
and the second one adds to (3/2)⋅ln(2).
This is a surprise. This definitely doesn't happen with finite sums,
and I wish it didn't happen for infinite sums...
So why is this?
The theorem says, that whether we can reorder the terms of a series or not,
depends on what type of convergent it is.
If a series is absolutely convergent, then what I just showed will not happen,
Then it can definitely be reordered, and this won't change the value of the sum.
If the original series was convergent,
after I change the order, it is still convergent, with the same sum.
But if the series is conditionally convergent,
then, well, reordering it may well change the value.
And it is actually worse!
If a series is conditionally convergent, then it is always possible to reorder it
to make it convergent to ANY number we want.
Choose what you want the value of the sum to be,
and there's a way to order it so that that's the value of the sum.
And there is also a way to reorder it to make it divergent in any way we want.
(Perhaps divergent to ∞, or oscillating.)
In other words, conditionally convergent series are bad.
This theorem is quite shocking.
It seems very powerful, and very surprising,
and it looks like it's going to be difficult to prove.
But surprisingly, it is not that difficult.
I'm going to try to explain what's bad about conditionally convergent series
first with an example,
and then we'll see that that works for all conditionally convergent series.
So I'm going to look more closely at this example:
the alternating harmonic series 1 - 1/2 + 1/3 - 1/4, and so on...
I'm going to point out that this example has three important properties.
First, if I look at the terms that I am adding and subtracting without the signs,
the limit of those terms is 0.
This will be true for any conditionally convergent series,
because it must be true for any convergent series.
Second, I'm going to look separately at the negative and positive terms.
If I look just at the negative terms: -1/2 - 1/4 - 1/6, and so on,
then they add up to -∞.
In this case,
one way to see that is if I take a common factor of -1/2 from these terms,
they give me the harmonic series itself, which I know to be ∞.
And separately, if I look just at the positive terms 1 + 1/3 + 1/5, and so on,
they add up to ∞.
One way to justify that is by comparing
the positive terms with the negative terms in absolute values;
and the positive terms are greater than the negative terms in absolute value,
so they also diverge to ∞.
And these three properties
turn out to be true for any conditionally convergent series.
Well it turns out that whenever I have a series
where the limit of the terms I'm adding is 0,
the positive terms add to ∞, and the negative terms add to -∞,
then I'm going to be able to reorder it to get any value of the sum I want.
And here is how!
As an example, I'm going to try to reorder the alternating harmonic series,
so that the sum is 7.
But there's nothing special about 7. It's a number I chose at random.
The same will work with any other number.
To do this, first, I'm going to add positive terms
(just positive terms, ignoring the negatives)
until a partial sum goes above 7.
I know this is possible because all the positive terms add up to ∞;
therefore, this would be possible to go as far as I need.
Then, I'm going to add negative terms until a partial sum goes below 7.
Again, that's possible because all the negative terms together add up to -∞.
And then, I go back to Step 1, and keep doing that
(adding positive terms until I go beyond 7,
and adding negative terms until I go below 7).
If I do this, it will definitely give me a reordering of the original series.
Why will this new series be converging to 7?
Now is when I use the third property.
Notice that the limit of the terms I am adding and subtracting is 0,
so every time I get to a partial sum that goes above or below 7,
they will be staying closer to 7,
because they will only go above or below by 1/n,
so they will be staying closer and closer to 7,
and thus, the limit of the sequence of partial sums is 7, that's it!
I love this theorem because it's shocking and surprising,
and yet the proof is delightfully simple.
In summary, absolutely convergent series are the good type of convergent.
They behave nicely (they have similar properties to finite sums),
and we can do with them many of the things we can do with finite sums.
But conditionally convergent series are treacherous.
They may be convergent, but don't trust them!
They misbehave, and they don't have the properties we would like them to have.