Functions and Series | World of Mathematics
Summing Infinite Series | by Lovkush Agarwal
Adding together a bunch of numbers is pretty easy, right? Well, adding together finitely many numbers is easy (in principle that is – most of us have enough trouble adding just a couple of numbers!), but what about adding together infinitely many numbers? This is precisely what an infinite series is. Here are some examples:
- 1 + 1 + 1 + 1 + 1 + …
- 1 – 1 + 1 – 1 + 1 – …
- 1 + 12 + 13 + 14 + 15 + …
- 1 – 12 + 13 – 14 + 15 – …
How might you try to calculate the value of these series? One of the first things one might try is calculating the sums term by term and seeing if any patterns emerge. If you try this (do have a try!), you might have thoughts along the following lines:
- “Does the series equal infinity?”
- “Calculating the series term by term I get alternating values: 1 then 0 then 1 then 0 then 1 then 0…, so the answer must be 1 or 0. Or maybe I have to choose the middle of these, so then the sum would equal 1/2."
- “Need a calculator now! Get (to two decimal places): 1, 1.5, 1.83, 2.08, 2.28, 2.45, … The keener people may keep going: The sum of the first hundred terms is 5.19, the first thousand is 7.49. These values are increasing very slowly so there must be a limit to how large the sum will become – there’s no way it can be infinite!”
- “Doing the calculations, I get 1, 0.5, 0.83, 0.58, 0.78, 0.62, … which looks like it will home in on a particular number around 0.7.”
Now that we have initial thoughts on these sums, we will go through each one and discuss what can be said in some detail.
Casually speaking, the answer is indeed infinity. Technically though, infinity is not a number but shorthand for “something that is not finite”, and since we want the value of a series to be a number, we conclude that this series has no meaningful value. When this happens, we say that the series diverges.
This is actually very easy to deal with by pairing the terms in the series:
|1 – 1||+||1 – 1||+||1 – 1||+||…|
So we conclude the series equals 0.
Not so fast! Unfortunately things are not as straightforward as they initially seem, since we could have paired up terms differently:
|1||– 1 + 1||– 1 + 1||– 1 + 1||–||…|
Now we conclude that the series equals 1! How is this possible?! What we have demonstrated is that we cannot manipulate infinite sums as freely as we can manipulate finite sums. Explicitly, you could group together terms in a finite sum however you want, and you will always get the same answer. However, as we have just seen, the value of an infinite sum can depend on the way we group together terms. And it does not end there. Let S denote the value of the series. So far we have shown that S = 0 and that S = 1. Now, lets write out the series twice, but with the second one being shifted to the right. Then add the two rows together:
Therefore S = 1/2! We can take this one step further, writing the series out thrice:
And therefore S = 1/3! Continuing in a similar fashion we can show that S equals 1/4, 1/5, 1/6, … We are not quite done yet though: we can actually show that S equals any rational whatsoever. For example, to show S = 4, we rearrange the series as follows:
|1 + 1 + 1 + 1||– 1 + 1||– 1 + 1||– 1 + 1||+||…|
“But where did the extra 1’s come from?”, you rightly ask. It looks like I just plucked them from thin air, but I actually just rearranged the 1’s and the –1’s in the series. In the original series (1 – 1 + 1 – 1 + 1 – 1 + …) we are adding up infinitely many 1’s and infinitely many–1’s. In the new version (1 + 1 + 1 + 1 – 1 + 1 – 1 + …) we are still adding up infinitely many 1’s and infinitely many –1’s. In other words, there are the same number of 1’s in both of the series, so the new series really is a rearrangement (as opposed to me illegally adding extra 1’s from nowhere).
So after doing all this funny business of pairing things up and rearranging, what do we conclude. Well, just as last time, we must concede and say that no meaningful value can be given to this series, i.e., that the series diverges.
As we observed earlier, this series increases incredibly slowly: the terms in the series get smaller and smaller, so we instinctively think there is a limit to how large the series can be. However, it turns out that this series is just like the first series: this sum (again, casually speaking) equals infinity!
How do we prove this? The idea is to first show that the series is bigger than 1, then that it is bigger than 112, then 2, then 212, then 3, and so on indefinitely. Before continuing, remind yourself that if 0 < m < n, then 1m > 1n, for example 15 > 18. We will be using this throughout the proof.
So to start off, by looking at the first two terms, we see that the series is bigger than 1 + 12.
Now we want to show the series is bigger than 2. We do this by looking at the next two terms: 13 + 14 > 14 + 14 = 24 = 12. So the sum of these two terms with the first two terms is bigger than 2.
Now we want to show the series is bigger than 212. This time we look at the next four terms: 15 + 16 + 17 + 18 > 18 + 18 + 18 + 18 = 48 = 12. So the sum of these four terms with the first four terms is bigger than 212.
Now we want to show the series is bigger than 3. This time we look at the next eight terms: 19 +110 + … + 116 > 116 + 116 + … + 116 = 816 = 12. So the sum of these eight terms together with the first eight terms is bigger than 3.
Take a moment here to think of what you would do to show that the series is bigger than 312. You should ask yourself: “At how many of the next terms should I look?”. Do not just guess – try to understand the pattern of choices that have been made in the first three steps. Here is how we continue:
This time we look at the next sixteen terms: 117 +118 + … + 132 > 132 + 132 + … + 132 = 1632 = 12.So the sum of these sixteen terms with the first sixteen terms is bigger than 312.
We continue in this manner, doubling the number of terms we sum up each time: to show that the series is bigger than 4 we look at the next 32 terms, bigger than 412 – the next 64 terms, and so on.
And there we have it! For any number, we have shown that the series is (eventually) bigger than that number, so, like for Series (i), we conclude that this series diverges. Of course, this series grows much slower than the first series: it takes about a quarter of a million terms for the series to get over 10, and more than 1043 terms to get over 100! This series actually has a name: it is known as the Harmonic Series. Take a look at its Wikipedia page for more information, including a physical interpretation of this result about a worm on a rubber band.
This series is known as the Alternating Harmonic Series. It behaves as described at the start: the sum is homing in on a number around 0.7. In fact, the number is exactly log(2), hence, this is the value of the sum. It is not easy to determine this, as it requires one to know the Taylor Series of the function log(x). Check Wikipedia for more information on Taylor series.
However, there is a word of warning here. Despite the fact we have a concrete value for this series, we still have to be careful with how we handle it because of the following result: Given any number x on the real line, you can rearrange this series so that the value of the series is x, not log(2).
This result is known as the Riemann Series Theorem, and it holds for many other series (but not all series). The argument uses the following three facts:
- The individual terms in the series get closer and closer to zero, e.g. the tenth term (–120) is closer to zero than the fifth term (15), the thousandth term (–11000) is even closer, and so on.
- The sum of the positive terms of the series, 1 + 13 + 15 + … is infinite.
- The sum of the negative terms of the series, 12 + 14 + 16 + … is infinite.
Note: Facts 2 and 3 can be proven from the fact the harmonic series is infinite.
So now suppose we are given an x; to make the argument more concrete, suppose x = 10. We are required to rearrange the series so that the sum of this rearrangement is 10. We do this as follows:
- At the start of the series, we use only positive terms.
- By Fact 2, there is a point where our sum will exceed 10. For the sake of clarity and concreteness, imagine it took a hundred of these terms to exceed 10. (In reality, we would need around a quarter of a million terms).
- Now start using negative terms. We do this because we (purposefully) overshot, and need to get back to 10.
- By Fact 3, there is a point where the sum will now be back below 10. For the sake of concreteness, imagine it took fifty negative terms to get back below 10.
- Keeping track: We have so far used up a hundred positive terms and fifty negative terms, and currently the sum is below 10.
- Now we want to get back above 10, so we start using positive terms again. By Fact 2, we know there will be a point where we do get back above 10; imagine it takes the next 150 positive terms to do this. (So in total, we have used up 250 positive terms.)
- Now we want to get back below 10, so we start using negative terms again. By Fact 3, we know there will be a point where we do get back below 10; imagine it takes the next 40 negative terms to do this. (So in total, we have used up 90 negative terms.)
- Keeping track: We first used 100 positive terms, then 50 negative terms, then 150 positive terms, then 40 negative terms, and the sum is currently below 10.
- Now we want to get back above 10, …
- Now we want to get back below 10, …
- We repeat this process, using up all the terms in the harmonic series.
So how does this rearranged version of the harmonic series behave? We will see that the term by term sum will oscillate back and forth around 10, so it looks like we have done the job. However, there is still one thing we need to show, which is that the term by term sum is also getting closer and closer to 10 (instead of oscillating between 9 and 11, for example).
As you may have guessed, this is where Fact 1 comes in. Fact 1 says that the terms (both positive and negative) are getting smaller and smaller. What this means is that the amount we overshoot by when we get over or below 10 must also be getting smaller and smaller. Remember, we start using negative as soon as the sum exceeds 10, and since the positive terms are getting smaller and smaller, the amount by which we exceeded 10 will get smaller. For example, the first time we exceed 10 we may have reached 10.2, but then the next time we will exceed 10 by a smaller amount, say we reach 10.15, and the next time we will exceed 10 by even less, say 10.08, and so on, getting closer and closer to 10.
One question that many of you may have is the following: do all series with only positive terms diverge? The answer is no. In fact, if you did A-Level maths you will have come across some examples, namely geometric series. For example:
1 + 12 + 14 + 18 + … + 12n + … = 2.
Even if you have not understood many of the details or arguments, you will hopefully have caught a glimpse of the fascinating and counter-intuitive nature of infinite series, and how bizarrely different they are from the finite sums we are used to dealing with on a day to day basis. Perhaps you may even be able to show some of your friends one or two of the little oddities you have picked up by reading this!
To end, I should admit I have been slightly dishonest. I said that we cannot give the second series any meaningful value, however that is not actually true. I have recently (and by pure coincidence) found out that it we can meaningfully give it a value, which, by the way, is 1/2. The reason I did not say this in the main text is because this is second hand knowledge that I do not understand and hence cannot explain. There is a book by G. H. Hardy all about divergent series (aptly named Divergent Series), so have a look if you are interested in finding out more!
More on functions and power series will be coming soon.