3: Sequence and series
- Page ID
- 89250
You have probably learned about Taylor polynomials 1 and, in particular, that
\begin{align*} e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} +E_n(x) \end{align*}
where \(E_n(x)\) is the error introduced when you approximate \(e^x\) by its Taylor polynomial of degree \(n\text{.}\) You may have even seen a formula for \(E_n(x)\text{.}\) We are now going to ask what happens as \(n\) goes to infinity? Does the error go to zero, giving an exact formula for \(e^x\text{?}\) We shall later see that it does and that
\begin{align*} e^x &=1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n=0}^\infty\frac{x^n}{n!} \end{align*}
At this point we haven't defined, or developed any understanding of, this infinite sum. How do we compute the sum of an infinite number of terms? Indeed, when does a sum of an infinite number of terms even make sense? Clearly we need to build up foundations to deal with these ideas. Along the way we shall also see other functions for which the corresponding error obeys \(\lim\limits_{n\rightarrow\infty}E_n(x)=0\) for some values of \(x\) and not for other values of \(x\text{.}\)
To motivate the next section, consider using the above formula with \(x=1\) to compute the number \(e\text{:}\)
\begin{align*} e &= 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots = \sum_{n=0}^\infty\frac{1}{n!} \end{align*}
As we stated above, we don't yet understand what to make of this infinite number of terms, but we might try to sneak up on it by thinking about what happens as we take more and more terms.
\begin{align*} \text{1 term}\phantom{s} && 1&=1\\ \text{2 terms} && 1+1&=2\\ \text{3 terms} && 1+1+\frac{1}{2}&=2.5\\ \text{4 terms} && 1+1+\frac{1}{2}+\frac{1}{6}&=2.666666\dots\\ \text{5 terms} && 1+1+\frac{1}{2}+\frac{1}{6} + \frac{1}{24}&=2.708333\dots\\ \text{6 terms} && 1+1+\frac{1}{2}+\frac{1}{6} + \frac{1}{24} + \frac{1}{120}&=2.716666\dots \end{align*}
By looking at the infinite sum in this way, we naturally obtain a sequence of numbers
\begin{gather*} \{\ 1\,,\,2\,,\,2.5\,,\,2.666666\,,\cdots,\,2.708333\,,\cdots,\, 2.716666\,,\cdots,\,\cdots\ \}. \end{gather*}
The key to understanding the original infinite sum is to understand the behaviour of this sequence of numbers — in particularly, what do the numbers do as we go further and further? Does it settle down 2 to a given limit?
- Now would be an excellent time to quickly read over your notes on the topic.
- You will notice a great deal of similarity between the results of the next section and “limits at infinity” which was covered last term.
- 3.1: Sequences
- A sequence is a list of infinitely many numbers with a specified order.
- 3.2: Series
- A series is a sum of infinitely many terms. You already have a lot of experience with series, though you might not realize it. When you write a number using its decimal expansion you are really expressing it as a series. Perhaps the simplest example of this is the decimal expansion of 1/3=0.333333...
- 3.3: Convergence Tests
- It is very common to encounter series for which it is difficult, or even virtually impossible, to determine the sum exactly.
- 3.4: Absolute and Conditional Convergence
- We have now seen examples of series that converge and of series that diverge. But we haven't really discussed how robust the convergence of series is — that is, can we tweak the coefficients in some way while leaving the convergence unchanged.
- 3.5: Power Series
- Let's return to the simple geometric series
- 3.6: Taylor Series
- Taylor polynomials provide a hierarchy of approximations to a given function f(x) near a given point a. Typically, the quality of these approximations improves as we move up the hierarchy.
- 3.7: Optional — Rational and irrational numbers
- In this optional section we shall use series techniques to look a little at rationality and irrationality of real numbers. We shall see the following results.