What are power series, and why are they worth learning about?
Power series are a way of expressing mathematical functions of [latex]x[/latex] as a sum of terms of powers of [latex]x[/latex]. We met an example in Section 5, when we expressed [latex]e^x[/latex] in the form [latex]e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^n}{n!}+\cdots[/latex]
In that expression, there is a constant term, a linear term (a term in [latex]x[/latex], that is, ‘[latex]x[/latex] to the power 1’), a term in [latex]x^2[/latex], a term in [latex]x^3[/latex], and so on. That’s a power series.
So what?
Well, often powers of [latex]x[/latex] are not too complicated to deal with mathematically. And real world situations can often be expressed mathematically as power series. Also, in certain circumstances, we can ignore many of the power terms because they have very small values. Then we leave ourselves an easier piece of maths to deal with. We say more about that in Section 10: Approximations.
To lead up to that, we’ll use this page to explain arithmetic and geometric series, then ‘infinite series’ (including convergence and divergence), and then ‘power series’. You might not want to know all that stuff if you arrived here just wanting to know about power series, but it does tell a conceptual story in a sensible order.
Arithmetic and Geometric Series
A ‘series’ is a sum of a sequence of numbers [latex]a_1+a_2+\cdots+a_n[/latex]. The numbers being added are called terms; [latex]a_n[/latex]is the [latex]n[/latex]th term. We could write this using the ‘sigma’ notation for summing: [latex]\sum_{i=1}^{n}a_i[/latex], which can be read “the sum of [latex]a_i[/latex] from [latex]i[/latex] equals [latex]1[/latex] to [latex]n[/latex]”. You may often hear the phrase ‘the sum of [latex]n[/latex] terms’; this is what it means.
The ‘[latex]i[/latex]’ in the previous paragraph is just a label. It is NOT the square root of -1 (see Section 06: Complex Numbers for details on that!). There are more things for scientists to describe than there are letters of the alphabet, so letters get re-used. If the terms of our series did contain complex numbers involving the number [latex]i[/latex], we should look for a different label to use in the summation: [latex]j[/latex] perhaps. In fact, electrical engineers often do it the other way round: [latex]i[/latex] is often reserved for ‘current’, so they use [latex]j[/latex] for the square root of -1 (!).
Everyday life is full of times you add numbers together, and thus full of series. A shopping bill is an example of a series, if not a very interesting one. For our purposes, series are more interesting if successive terms are linked by a rule, unlike the random nature of shopping bills. All the series we explore on this page will be of this more interesting type. So the series [latex]1+3+5+\cdots+27[/latex] is governed by the rule ‘each term is the previous term plus 2’. The ellipsis ([latex]\cdots[/latex]) signifies that we should carry on going with the rule until we get to 27. A series for which the rule to generate it is “add a fixed amount more than the previous term” is called an arithmetic series.
In contrast, a geometric series is a series where the rule between terms is ‘multiply by a certain amount’, rather than ‘add a certain amount’.
Suppose the number of bacteria in a culture doubles every hour. One way this could come about would be “every bacterium divides in two every hour; no bacteria die”. Then the number of bacteria is successive hours is 1, 2, 4, 8, 16, 32, 64… The rule between the terms of the sequence is “multiply by 2”. Strictly speaking, this list is a geometric sequence, not a geometric series, because we are just listing the terms, not adding them.
Geometric sequences and series can have amazing properties, and can take some getting used to. Here are two examples:
- Let’s say a piece of paper is 0.1 mm thick. If we fold it, leaving no space between the two ‘half-sheets’, the doubled piece of paper will be 0.2 mm thick. Fold it again, and the thickness of the stack is now 0.4 mm. The thickness doubles with every fold. How high would the stack of paper be if you could fold it 42 times? It would reach to the moon.
- The second example concerns a famous story, reproduced here as it was worded in forbes.com. ‘As the story goes, when chess was presented to a great king, the king offered the inventor any reward that he wanted. The inventor asked that a single grain of rice be placed on the first square of the chessboard. Then two grains on the second square, four grains on the third, and so on. Doubling each time. The king, baffled by such a small price for a wonderful game, immediately agreed, and ordered the treasurer to pay the agreed upon sum. A week later, the inventor went before the king and asked why he had not received his reward. The king, outraged that the treasurer had disobeyed him, immediately summoned him and demanded to know why the inventor had not been paid. The treasurer explained that the sum could not be paid – by the time you got even halfway through the chessboard, the amount of grain required was more than the entire kingdom possessed. The king took in this information and thought for a while. Then he did the only rational thing a king could do in those circumstances. He had the inventor killed, as an object lesson in the perils of trying to outwit the king.’ In fact, had the end of the chess board been reached, the rice would form a layer that covered the entire land surface of the earth.
Note that the first of those two examples involved a ‘sequence’, whereas the second involved a ‘series’ – the king’s debt involved all the rice on all the squares added together, not just the rice on the final square.
Infinite series
An infinite series is a series that goes on forever: [latex]a_1+a_2+\cdots+a_n+\cdots[/latex]. The ellipsis at the end of the expression signifies that the series never stops. When the summation notation ([latex]\Sigma[/latex]) is used, a series can be identified as infinite by the use of the infinity sign ([latex]\infty[/latex]) as the designated last term: [latex]\sum_{i=1}^{\infty}a_i[/latex].
Convergent and divergent series
A finite (not infinite) series has a finite sum. That is, if you add a finite amount of numbers together, you will get a finite answer, however large or small. For example, the first series we looked at, [latex]1+3+5 +\cdots+27[/latex], has the sum 196. The total number of grains of rice owed by the king to the inventor is [latex]1.844\times10^{19}[/latex] (which is a lot, but it’s not infinite).
It’s reasonable to ask now ‘can an infinite series have a finite sum?’
If we carry on the [latex]1+3+5+\cdots+27[/latex] series forever (rather than stopping at 27), the infinite series will not have a finite sum, because every term we add contributes more to the sum. If we carry on forever, the sum will be infinite. Likewise for the grains of rice on an infinitely large chessboard. These sums are described as divergent – the sum of [latex]n[/latex] terms diverges (keeps getting further away from the previous sum as [latex]n[/latex] gets larger).
In contrast, look at the sum [latex]1+\frac{1}{2}+\frac{1}{4}+\cdots[/latex], where each term is half that before. We could write this sum as [latex]\sum_{i=1}^{\infty}\frac{1}{2^{i-1}}[/latex]. This sort of notation is common, so we will spend a little time decoding it in the video below. If the notation is obvious to you, maybe skip the video.
This particular series can be shown easily in diagram form. In the picture below, the full-length blue bar is ‘worth 1’. And the dotted rectangle that includes that bar is then ‘worth 2’.
You can see from the picture above that:
- The sum of the first two terms is [latex]1+\frac{1}{2}=1.5[/latex]
- The sum of the first three terms is [latex]1+\frac{1}{2}+\frac{1}{4}=1.75[/latex]
- The sum of the first four terms is [latex]1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=1.875[/latex]
As [latex]n[/latex] increases, the sum of the series approaches the value 2 (notice how close to a value of 2 we get even with a small number of terms – look at the diagram for [latex]n=8[/latex], for example). We can say that the sum of the infinite series is 2. Or
[latex]\sum_{i=1}^{\infty}\frac{1}{2^{i-1}}=2[/latex]
There is an easy formula to find the infinite sum (and indeed another one for the sum of [latex]n[/latex] terms) of any geometric series, but that is not the focus of this page. If interested, you can find out, say on the Wikipedia page for geometric series.
The result above is interesting in the context of this page – this infinite sum does have a finite value, unlike the divergent series we saw earlier. It is called a convergent series. Unlike our examples of divergent series, each term is smaller than the previous term – by the same ratio each time because it is a geometric series. (For non-geometric series, ‘the terms get smaller’ is not a sufficient condition for the series to be convergent, and we won’t worry here about how you tell whether an infinite series is convergent or not).
Let’s have another look at the sum of [latex]n[/latex] terms of the series
[latex]\sum_{i=1}^{n}\frac{1}{2^{i-1}}=1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n-1}}[/latex]
Here they are, in the following table.
[table id=7 /]
Look how quickly the sum of terms approaches the value 2. The value 2 is the sum of the infinite series (as if the table carried on forever downwards). After only 7 terms, we are less than 1 % from the value for an infinity of terms!
We won’t meet geometric series much in the rest of this site, but we wanted to show you this idea that some series have almost all the magnitude contained in the first few terms. The first seven terms contain over 99 % of the value of the series, so ignoring all the other terms will lead to an error of less than 1 %. This is an important idea in science – by ignoring later terms and just concentrating on the few that have the biggest impact, we can often make problems easier (although of course it’s not necessary for a geometric series – because its Wikipedia page tells you how to find the infinite sum really easily!). We discuss this in Section 10 (Approximations).
There is a story about the mathematician John von Neumann that is too irresistible to omit here. You can find a version of it at pleacher.com…
Power series
A power series is a series in which each of the terms is a different power of an unknown, say [latex]x[/latex], multiplied by a coefficient.
That is,
[latex]f(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdots+a_nx^n[/latex]
[latex]a_n[/latex] is the ‘coefficient’ of [latex]x^n[/latex], that is, the number that [latex]x^n[/latex] needs to be multiplied by, to make the whole thing work. The coefficients are labelled so that their number (in the subscript) is equal to the power of [latex]x[/latex] it belongs to. (Be careful! That means that the first one has the label ‘zero’, whereas in all the previous series on this page, the first term was labelled 1, for ‘first’!) The values of the coefficients (along with the values of [latex]x[/latex]) determine which terms are most important in the series.
There is a piece of mathematics called Taylor’s theorem, which is very useful. It says that ANY function can be expressed as an infinite series of powers of [latex]x[/latex], as above, provided it meets certain conditions. We aren’t going to worry about those conditions – we are trying to give an insight into power series; if you need a working knowledge of Taylor’s theorem, an undergraduate maths book might be a good starting point!
It turns out that [latex]\sin{x}[/latex], [latex]\cos{x}[/latex], and [latex]e^x[/latex] can all be expressed as power series in [latex]x[/latex] (also sometimes called ‘series expansions’). And that is of interest to science students, because many physical phenomena are described using trigonometric or exponential functions, or both. To find out how to express these functions as a power series in the video below, we rather optimistically assume they can be expressed this way, and then find the coefficients that make it work. If we can’t find the coefficients, then maybe we have a function that it doesn’t work for.
Before you watch the following video, you need to be aware of a ‘factorial’, represented by an exclamation mark. So [latex]n![/latex] is ‘[latex]n[/latex] factorial’. It means multiply the whole number [latex]n[/latex] (it has to be a whole number) by the number one less, and again by one less than that, and so on until you get to 1. So, for example 5! means 5 x 4 x 3 x 2 x 1, so that 5! has the value 120. Likewise, 6! = 720.
Also, if you’ve never seen the method in the video before, don’t worry. We’re not saying that you should have been able to work it out for yourself. We just hope you can follow it, so that you can be confident the conclusions we arrive at are true.
In the video, we promised a reason for 0! being defined as 1. You can find a clear explanation here.
To summarise the video:
[latex]e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^n}{n!}+\cdots[/latex]
Or
[latex]e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}[/latex]
This is [latex]e^x[/latex] expanded as a power series.
Next, we’ll use the same technique to expand [latex]\sin{x}[/latex] as a power series. To understand this video, you’ll also need to be able to differentiate sine and cosine functions – see Section 02 for details if you need a reminder on what this means.
To summarise the video:
[latex]\sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots[/latex]
Or
[latex]\sin{x}=\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k+1}}{(2k+1)!}[/latex]
Finally, although [latex]\cos{x}[/latex] wasn’t treated in the video, you can find its power expansion in just the same way. You might not be surprised that it is very similar to the expansion of [latex]\sin{x}[/latex] (and if you are really keen, you might like to work it out yourself, adapting the steps from the video). If you don’t want to go through all that, but you have remembered that [latex]\cos{x}[/latex] is the derivative of [latex]\sin{x}[/latex], then you could differentiate each term in the series expansion (above) for [latex]\sin{x}[/latex] instead…
And for those who don’t want to work it out themselves, here it is:
[latex]\cos{x}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots[/latex]
Or
[latex]\cos{x}=\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k}}{(2k)!}[/latex]
As a reminder of the behaviour of the exponential function, you might like to return to Section 5 – What ‘exponential’ means. And if you have noticed that the power expansions for sine and exponential functions looks quite similar, then you are anticipating Section 12, which will cover Euler’s formula and why it is so useful to scientists.
Another aside: you might have noticed that we summed the series for [latex]e^x[/latex] with the label [latex]n[/latex] and the series for [latex]\sin{x}[/latex] with the label [latex]k[/latex]. Why the difference? Well, [latex]n[/latex] denotes the power of [latex]x[/latex]. We can use this label for [latex]e^x[/latex] because all powers of [latex]x[/latex] are present in the expansion. But with [latex]\sin{x}[/latex], only the odd powers are present, and we use all values of the the variable [latex]k[/latex] to generate the odd values of [latex]n[/latex], through the expression [latex]2k+1[/latex].
Truncating power series to make approximations
Here’s a shock! Your electronic calculator, and computer, can’t calculate sines, cosines and exponentials from the geometry of the situation. All they can do is add, subtract, multiply and divide. So when you ask your calculator to work out [latex]\sin{40^{\circ}}[/latex] and it (almost instantly) returns the answer 0.642787609, you should be pretty impressed with it. Because it has calculated the answer from the power series! Which requires a lot of multiplying and adding.
You might object that it would require an infinite amount of adding because the power expansions of these functions are infinite series! But remember we showed you a geometric series in which the vast majority of the value of the sum was contained in the first few terms? Well, often, something similar applies to power series too. So your calculator can turn the infinite series into a finite series by ‘truncating’ it – cutting it short after a suitable number of terms. Where should your calculator truncate it? Well, as soon as the terms being ignored make no difference to the number of decimal places on the display. You want a calculator with more decimal places? Then it will need to calculate more terms in the series as it computes sines and the like.
This idea of truncating power series to make approximations is very powerful. We will revisit it in Section 10 (Approximations).
And finally…
We have covered a lot of ground on this page. So here’s a reminder of the key message:
“You can express exponential and trigonometric functions (and many others) as infinite power series, and that can be useful because powers of a variable are easy(ish!) to work with.”
Link to Section 08 – Work links forces with energy
On the basis that this is Section 7, it would make sense to go to Section 8
Link to Section 05 – What ‘exponential’ means
Or you could go back here, to remind yourself about exponential functions
Link to Section 10 – Approximations (coming soon)
This page will look at how to make life easier in physics when the full problem is too hard, by among other things, applying your knowledge of truncating power series
Link to Section 09 – Conservation laws (coming soon)
This page will set up ideas of conservation of energy…
Home page
And you can always go back here…