What are power series used for? (a reference request)

Solution 1:

At the moment I can't think of a good reference for what you want (and I consider myself fairly knowledgeable on literature matters such as this), so what I suggest is something I've done many times over the years when I was in a similar situation. Jot down all the applications you can think of in a notepad. Then look through all the books on your bookshelf for additional examples and ideas. Look at calculus texts, advanced calculus texts, real analysis texts, upper level physics texts, upper level engineering texts --- anything that you think has a chance of containing something useful. (Physics and engineering texts often contain nice real world examples you can use.) Make sure to look at the exercises, because often this is where you'll see something you don't already know about. Repeat the process at your university library. You may need to make two or three trips there, unless you can spare several hours one afternoon and have the patience to do so.

Once you have started such a list, keep all your notes for it in a separate folder. This is important because over time you will stumble upon other applications, so when this happens, all you have to do is jot it down (or print it out if it's something you see online, like in math stackexchange which has "power series" as a tag) and slip the sheet into your folder for power series stuff.

All this may seem like a lot of work just for power series applications, but the benefits can go beyond this. Do the same for any of the one or two dozen other topics you'd like to have examples of. I made many such folders when I was teaching in the 1980s and 1990s, and I often came across (by accident) interesting new things to add to existing folders when I was initially "researching" a new topic. In the 2000s I began posting some of these for other people to use (mostly in sci.math, followed by the Math Forum group AP-calculus). For example, here is a list of some things related to rationalizing the denominator that I posted in 12 June 2001. However, I don't believe I've posted such a list for power series, although a (likely tedious) search through my posts over the years should bring up a lot applications.

Although my main point here was to explain a useful long-term teaching strategy, I'll list some of the applications (and supplementary topics) I can think of off-hand or find on a few of my old tests that I happen to have access to where I'm at.

You can use geometric series expansions (along with various algebraic manipulations) to obtain the power series expansions and intervals of convergence for many rational functions, such as $\frac{2x}{4+x}$ (which equals $\frac{1}{2}x$ times $\frac{1}{1-r},$ where $r = -\frac{x}{4})$ and $\frac{5x^3}{4 - x^3}$ (which equals $\frac{5}{4}x^3$ times $\frac{1}{1-r},$ where $r = \frac{1}{4}x^{3}).$

Power series expansions can sometimes be used to easily evaluate limits that are difficult by other methods (L'Hopital, etc.), and I've seen many examples of these in math stackexchange. An example from an old test of mine is the following:

Evaluate the following limit by the Taylor series method:

$$ \lim_{x \rightarrow 0} \, \left[ \frac{1 - \cos\left(\pi x^2\right)}{x^{2}\sin\left(ex^{2}\right)} \right] $$

Students were told to memorize the expansions of a few basic functions, which included those needed above, so a solution required no more work than

$$\frac{1 \; - \; \left[1 - \frac{1}{2}\left(\pi x^{2}\right)^{2} + \cdots \right]}{x^{2} \cdot \left[\left(ex^{2}\right) - \cdots \right]} \;\;\; = \;\;\; \frac{\frac{1}{2}{\pi}^{2}x^4 + \cdots}{ex^4 - \cdots} \;\;\; \rightarrow \;\;\; \frac{{\pi}^{2}}{2e} $$

Power series expansions can be used to approximate the values of definite integrals, and a common example is the error integral (integrand is $e^{-x^2})$ because this leads to an alternating series (even when $x$ is negative), and so the error can be easily estimated. Sometimes I've asked a question like this for an integral they can evaluate by hand, so that they can check their work, such as the following problem also from an old test. [To find the 5th order Taylor polynomial, students only needed to substitute $u = -2x^2$ into the expansion of $e^u$ and then multiply term-by-term by $x.$ Note also that this problem is additionally intended to drive home the distinction between an estimate for an error and the error itself. At least one problem nearly identical to this would have been previously worked in class, and at least two or three would have been given for homework before the test.]

(a) Write down the 5th order Taylor polynomial about $x=0$ for the function $f(x) = xe^{-2x^{2}}.$

(b) Use your answer in (a) to approximate the value of $\; \int_{0}^{0.5} f(x) \, dx.$

(c) Use the alternating series test to obtain an upper bound on the error in the approximation you made in (b).

(d) Find the exact value of this definite integral (let $u = -2x^{2})$ and use this exact value to find the exact error of the approximation you found in (b). You should find that the exact error is $\leq$ your upper bound for the error (your answer to (c)).

Here is a fun apparent paradox from one of my "supplementary problems" homework assignments:

Using geometric series ideas, it is easy to see that $\frac{x}{1-x} = x + x^2 + x^3 + \cdots$ and $\frac{x}{x-1} = \frac{1}{1 - x^{-1}} = 1 + x^{-1} + x^{-2} + \cdots.$ Hence, $\frac{x}{1-x} + \frac{x}{x-1} = \cdots + x^{-2} + x^{-1} + 1 + x + x^2 + \cdots = \sum_{n = -\infty}^{\infty}x^{n}.$ However, by high school algebra we get $\frac{x}{1-x} + \frac{x}{x-1} =0.$ Explain this apparent paradox by finding the interval of convergence for each of the series being added.

Here are 4 more such homework "supplementary problems":

1(a) Find a closed form expression for $\sum_{n=1}^{\infty}e^{nx}$ and give the interval of convergence.

1(b) Find the value of $x$ such that $\sum_{n=1}^{\infty}e^{nx} = 42.$

1(c) Find the value of $x$ such that $\sum_{n=1}^{\infty}e^{nx} = e.$

1(d) What is the range of the function $f(x) = \sum_{n=1}^{\infty}e^{nx}?$

$\;$

2(a) Express $(1-x)^{-1}$ in the form $\sum_{n=0}^{\infty}a_{n}x^{n}.$ For which values of $x$ is this series valid?

2(b) Evaluate $\int_0^{\frac{1}{2}}\left[\sum_{n=0}^\infty a_n x^n\right]\,dx$ for the series you found in (a) by integrating term-by-term. Your answer should be a numerical infinite series having a fairly straightforward pattern.

2(c) Evaluate $\int_0^{\frac{1}{2}}\left(\frac{1}{1-x}\right)dx$ in exact form. Note: If you've done (b) and (c) correctly, you should be able to express $\ln 2$ as the sum of an infinite series having a simple pattern. (An example where $\ln 2$ is expressed as the sum of an infinite series having a not so simple pattern is the decimal expansion of $\ln 2.)$

$\;$

3(a) Epress $(1+x^2)^{-1}$ in the form $\sum_{n=0}^\infty a_n x^n.$ For which values of $x$ is this series valid?

3(b) Evaluate $\int_0^1 \left[\sum_{n=0}^\infty a_n x^n\right]\,dx$ for the series you found in (a) by integrating term-by-term. Your answer should be a numerical infinite series having a fairly straightforward pattern.

3(c) Evaluate $\int_{0}^{1}\left(\frac{1}{1+x^{2}}\right)dx$ in exact form. The antiderivative of the integrand will involve the arctangent function and the numerical evaluations involve values of the arctangent that you should know. Note: If you've done (b) and (c) correctly, you should be able to express $\pi$ as the sum of an infinite series having a simple pattern. (An example where $\pi$ is expressed as the sum of an infinite series having a not so simple pattern is the decimal expansion of $\pi .)$

$\;$

4(a) Express $(1-x)^{-1}$ in the form $\sum_{n=0}^{\infty}a_{n}x^{n}.$ For which values of $x$ is this series valid?

4(b) Evaluate $\frac{d}{dx}\left(\sum_{n=0}^{\infty}a_{n}x^{n}\right)$ for the series you found in (a) by differentiating term-by-term.

4(c) Evaluate $\frac{d}{dx}(1-x)^{-1}$ in closed form using short-cut differentiation rules.

4(d) By evaluating your answer to (b) and (c) for $x=\frac{1}{2},$ find the value of $\sum_{n=0}^{\infty}\frac{n+1}{2^n}.$

4(e) By subtracting term-by-term $\sum_{n=0}^{\infty}\frac{1}{2^n}$ (whose value you should know) from $\sum_{n=0}^{\infty}\frac{n+1}{2^n},$ find an exact expression for the value of $\sum_{n=0}^{\infty}\frac{n}{2^n}.$

4(f) Find a closed form value for $\sum_{n=0}^{\infty}\frac{an+b}{2^n}$ in terms of the constants $a$ and $b.$