When Professor Hamman first brought up the topic of series, and pointed out how not all series need add up to one, he reminded me of something I had heard a few years ago. When I was in Calculus II, and Prof. Wheatley was explaining series, he said that he could take a series and have it add up to whatever he wanted it to. However, he could only do this with certain series. Prof. Wheatley's argument was thus: Because we are adding up all of the terms of the series, we should be able to add them up in any order we want to and have the correct value.
So let's look at the series {1/2, 1/3, 1/4, 1/5, ...} again. As Prof. Hamman showed, this series should add up to infinity. So let's change the order around some; let's move the 1/4 in front of the one third. We now have {1/2, 1/4, 1/3, 1/5, 1/6, 1/7, ...}. As you can see, the sum should still be the same. However, let's move the 1/8 in front of the 1/3 also. And let's do the same thing with all of the powers of two we can find. The series now looks like {1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...}. Intuition would say that this should add up to the same number, but Prof. Hamman showed that this series adds up to one
So what went wrong here? If we count the series in that order, we will only ever be counting powers of two. In doing so, we will be skipping a large part of the series, which will never get counted. This is like Hilbert's Hotel, where we had a bus with an infinite number of people onboard. If we gave the people on the bus rooms in normal order, everyone would get their own room. However, if we were to try giving all of the odd-numbered people rooms first, not everyone would get a room, as we would have no end to the number of odd-numbered people on the bus.
So what does this mean? Just like before, if we take a set with a finite number of terms, no matter what order we count in, we will count them all with no exception. However, if we take an infinite set, we can count in some order so that we can skip terms and never count them, even if we do not stop counting. This also relates to the idea of one-to-one correspondence. Just as we can take an infinite set and form a one-to-one correspondence with a subset of itself, we can do the same with the terms of a series. The important thing to note here, though, is that doing so with a series changes the series, changing what it will add up to. So, in the end, we cannot have a series add up to whatever we want it to without changing the series itself. Sorry if I got your hopes up.
So let's look at the series {1/2, 1/3, 1/4, 1/5, ...} again. As Prof. Hamman showed, this series should add up to infinity. So let's change the order around some; let's move the 1/4 in front of the one third. We now have {1/2, 1/4, 1/3, 1/5, 1/6, 1/7, ...}. As you can see, the sum should still be the same. However, let's move the 1/8 in front of the 1/3 also. And let's do the same thing with all of the powers of two we can find. The series now looks like {1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...}. Intuition would say that this should add up to the same number, but Prof. Hamman showed that this series adds up to one
So what went wrong here? If we count the series in that order, we will only ever be counting powers of two. In doing so, we will be skipping a large part of the series, which will never get counted. This is like Hilbert's Hotel, where we had a bus with an infinite number of people onboard. If we gave the people on the bus rooms in normal order, everyone would get their own room. However, if we were to try giving all of the odd-numbered people rooms first, not everyone would get a room, as we would have no end to the number of odd-numbered people on the bus.
So what does this mean? Just like before, if we take a set with a finite number of terms, no matter what order we count in, we will count them all with no exception. However, if we take an infinite set, we can count in some order so that we can skip terms and never count them, even if we do not stop counting. This also relates to the idea of one-to-one correspondence. Just as we can take an infinite set and form a one-to-one correspondence with a subset of itself, we can do the same with the terms of a series. The important thing to note here, though, is that doing so with a series changes the series, changing what it will add up to. So, in the end, we cannot have a series add up to whatever we want it to without changing the series itself. Sorry if I got your hopes up.
Great job on the explanation! I totally "get" what you're saying. To help me understand it better, I think of it as a recipe. If I have a certain number/infinite number of ingredients and need to make a cake I can do that. If I want to make something else like bread or a pie with those same ingredients I can do that too. The same concept applies if I change the order that I add ingredients; it can change the entire outcome.
ReplyDeleteHi Julian,
ReplyDeleteYou are right. I'm doing sequential equations now in my calculus class and it's almost the same explanation as to what we are discussing now. I realized every term we use doesn't have a finite set because your series keep on adding up. In each sequence you can come out with your own formula to define your term, to give you whatever order you want to have. Switching the series in different order doesn't define your term all the time, some will work and others wouldn't work. There are different types of series and each of them fit in their own terms. I agree to the fact that infinity cannot be equal to infinity or add up to infinity. Each of them depends on their own terms by either adding them up or taking some out.
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i guess i'm just still confused how some infinities can total 1 and others can't solely by the way they're arranged. I see it like if you change an order it doesn't change the number of how many there are. Despite my confusion, I still grasped what we learned in class (or at least I think did). This post does help in easing my confusion, thanks Julian. I just didn't really see the point in adding the infinities to make it one. I feel like there was no defined attribute from all this. What I see is more of people manipulating something to get the result desired. With math, there is a lot of that. I just don't regard the ability to sequence something as fact of its existence. Like, do we create problems and patterns that don't really exist? If we (humans) were not present would these patterns exist? I think of it like the tree falling scenario where there is no one to hear it.
ReplyDeleteTracy your comparison to cooking does make this concept more comprehensible. However, i just see it as manipulation. And with this manipulation, one creates a new doorway.
I think what I meant in my post was that we can't manipulate the sequence to get the desired result. We can't have one sequence add up to two different things. If we try doing as I said, we just end up with two different sequences. So yes, we can pick a number, and find a sequence that adds up to that number, but we can't pick a sequence and then choose what we want it to add up to.
ReplyDelete