In a calculus course I just completed I struggled to understand the Taylor and Maclaurin series. I got the basics but not some of the harder concepts. Could anyone point me in the right direction? Books, tutorials, etc?
Many thanks.
[Math] Learning Taylor/Maclaurin Series
sequences-and-series
Related Solutions
Thanks for all the responses so far. I thought I would summarize them for anyone else who might be interested in this question. Rather than continuing to update the summary in the original question, though, it seems better from an organizational standpoint for the summary to appear in a community wiki answer than I then accept so that it appears at the top of the list of answers.
Others seem to agree that focusing on Taylor series is the right approach. See the answers and comments for justifications.
Examples of the use of infinite series
a. General: Zeno's paradoxes
b. Physics: Using the first order Taylor approximation $\sin \theta \approx \theta$ in solving the pendulum differential equation
c. Chemistry: Extending the ideal gas law to apply to high pressure and low temperature situations
d. Economics: Calculating fiscal multipliers involves geometric series
e. Computer science, 1: Several uses for generating functions (see examples by robinhoode and Raphael)
f. Computer science, 2: Taylor series are involved in the error analysis of some numerical methods, such as Newton-Raphson and Simpson's rule.
g. Mathematics, 1: Taylor series show that calculations involving functions like $e^x$ and $\sin x$ can all be computed using just addition, subtraction, multiplication, and division.
h. Mathematics, 2: Power series, and Euler products in number theory in particular, as most people find number theory intrinsically interesting whether they have the background or not
i. Mathematics, 3: Taylor series can be used to solve differential equations. (Often students will have seen a brief introduction to differential equations earlier in the course.)
j. Mathematics, 4: There are infinite series expressions for interesting constants such as $\pi$ and $e$. Also, any nonterminating decimal representation of a real number is an infinite series.
k. Mathematics, 5: Using Taylor polynomials to approximate integrands in definite integrals. (This fits well in a course like Calculus II that spends a lot of time on the integral.)
Links to other resources
a. Graphs of Taylor polynomials converging to a function (Brandon Carter's answer)
b. Graphs of Taylor polynomials in the complex plane (Hans Lundmark's comment on Brandon's answer)
Other
a. Emphasize similarity of series and improper integrals
b. Emphasize that convergence must be dealt with carefully. Use geometric series as a way to introduce convergence issues. Mention that even Euler was not always good about handling convergence.
You can just use the series expansion: $$\cos u = 1 - \frac{u^2}{2!} + \frac{u^4}{4!} - \cdots$$ then substitute $u$ for $\frac{x^2}{14}$, and finally multiply everything for $15x$. Probably it's the easiest way. Ok?
Best Answer
I think it's great that you're seeking a deeper understanding of an important topic-- even after the class is over!
If you haven't done so, go back to your calculus text and carefully read the sections on Taylor and MacLaurin series. (I know math textbooks often seem obtuse and unhelpful, but it's a valuable skill to be able to read slowly and learn with just the text as your guide.) Pay close attention to the examples, because they are acutely chosen to help you see the ideas and connections you need. If there's a gap you can't fill in, ask! --the more detailed your question, the more direct our help will be.
My best guess is that solving some problems by hand would make you more comfortable working with power series. Here are some problems to try. (It may take a lot of writing, but as long as you have one idea of what to do next, try it! There's a lot to keep track of with power series, and practice is important.)
Prove, by expanding as MacLaurin series, that $e^{ix} = \cos x + i \sin x$.
Start with the MacLaurin series for $(\cos x)$ and $(\sin x)$, square them, then add to show that $\cos^2 x + \sin^2 x = 1$. (To start, just work up to $x^4$, but go further if you can.)
Start with the MacLaurin series for $e^x$ and $\ln (1-x)$ and show that $e^{\ln (1-x)} = 1-x$. (Again, just work up to $x^4$ or so. You could also try the same thing with $e^{-\ln(1-x)}$.)
Prove, using MacLaurin series that $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx} \sin x = \cos x$, and $\frac{d}{dx} \cos x = - \sin x$. (For these, don't stop at $x^4$. You can use summation notation to work with all terms, from 0 to infinity, at the same time.)
Write the MacLaurin series for $\frac{1}{1-x}$, then take the integral of the series. (What do you get?) Take the derivative of the series. (What do you get?)
Find the MacLaurin series for $(\arctan x)$ by taking its derivative, finding the MacLaurin series expansion, then integrating that expansion.
I don't have other references for you, but power series expansions can be used in solving limits (replacing L'Hospital's rule), in differential equations, in combinatorics (generating functions), and in complex analysis. If any of those topics piques your interest, I'm sure people here or in your department can give some references to check out.