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We call this ''R'' the '''radius of convergence''' and in the case of (i) we say ''R''=0, and in the case of (ii) we say ''R''=<math>\infty</math>. | We call this ''R'' the '''radius of convergence''' and in the case of (i) we say ''R''=0, and in the case of (ii) we say ''R''=<math>\infty</math>. | ||
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| + | Let's say we have some function that isn't a particularly nice function. And by not "nice", I mean something like <math>f(x)=e^x \text{or} ln(x)</math> because if I asked you what <math>e^3.81</math> was you would have to use a calculator to find an approximate answer. But what if there was a way to rewrite a "nasty" function as a polynomial? Polynomials are generally easier to compute so it would be great if we could represent difficult functions as polynomials or power series (just infinite polynomials, remember). As it turns out, we can represent some functions as power series and if a function can be represented as a power series, we can find it using a '''Taylor Series''. | ||
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==TOPIC 2== | ==TOPIC 2== | ||
Revision as of 04:15, 22 May 2014
Contents
A Guide to Taylor and Maclaurin Series
by: Kathryn Marsh, proud Member of the Math Squad.
keyword: taylor series, maclaurin series
INTRODUCTION The purpose of this tutorial is to give an overview of Taylor and Maclaurin Series; what they are, how to derive them, and a few applications. This is meant to be a guide to UNDERSTANDING them and finding Taylor Series expansions of functions, not just being able to solve problems on your homework because math is a lot more fun that way :).
Contents - The Almighty Power Series - What's a Taylor Series? - Why is this useful again? - References
The Almighty Power Series
Before we get too deep into the magic of Taylor Series, we need to start with a firm understanding of the power series. So let's take a look at what a power series is.
A power series is just a polynomial that may or may not have a finite degree. Formally, this is anything of the form
$ \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + ... $
where the $ \ c_n $'s can be any constant and x is a variable.
Sometimes it is more useful to write a power series in another form, called a power series centered at a or a power series about a which we write $ \sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1 (x-a) + c_2 (x-a)^2 + c_3 (x-a)^3 + ... $
Let's say we have the following function.
$ \ f(x)=c_0 + c_1 (x-a) + c_2 (x-a)^2 + c_3 (x-a)^3 + ... $
Now as soon as we choose a specific x to plug into our function, we have an infinite series which may or may not converge. The domain for this function is the set of all x's for which the series converges.
How would we go about actually finding which x values are in the domain? To do this, we treat x as a number and proceed the same as if we had any other infinite series. Recall that we have several tests at our disposal for finding whether a series converges or not, namely, the integral test, the comparison and limit comparison tests, alternating series test, the p-series test, the ratio test, and the root test. In general, the ratio test is the most useful for these kinds of series but it may be that another test would also work.
Now, if we apply the ratio test to our function we can get three possible outcomes:
(i) The series only converges when x=a.
(ii) The series converges for all x.
(iii) There is some positive number R such that the series converges if $ \ |x-a|<R $ and diverges if $ \ |x-a|>R $.
We call this R the radius of convergence and in the case of (i) we say R=0, and in the case of (ii) we say R=$ \infty $.
What's a Taylor Series?
Let's say we have some function that isn't a particularly nice function. And by not "nice", I mean something like $ f(x)=e^x \text{or} ln(x) $ because if I asked you what $ e^3.81 $ was you would have to use a calculator to find an approximate answer. But what if there was a way to rewrite a "nasty" function as a polynomial? Polynomials are generally easier to compute so it would be great if we could represent difficult functions as polynomials or power series (just infinite polynomials, remember). As it turns out, we can represent some functions as power series and if a function can be represented as a power series, we can find it using a 'Taylor Series.
TOPIC 2
Lorem Ipsum [1] is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.
REFERENCES
[1] "Loream Ipsum" <http://www.lipsum.com/>.
Questions and comments
If you have any questions, comments, etc. please, please please post them below:
- Comment / question 1
- Comment / question 2
The Spring 2013 Math Squad 2013 was supported by an anonymous gift to Project Rhea. If you enjoyed reading these tutorials, please help Rhea "help students learn" with a donation to this project. Your contribution is greatly appreciated.
