Learning Taylor/Maclaurin Series

Question

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.

Answer 1

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.