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Section3.5Power Series

Let's return to the simple geometric series

\begin{gather*} \sum_{n=0}^\infty x^n \end{gather*}

where \(x\) is some real number. As we have seen (back in Example 3.2.4 and Lemma 3.2.5), for \(|x| \lt 1\) this series converges to a limit, that varies with \(x\text{,}\) while for \(|x|\geq 1\) the series diverges. Consequently we can consider this series to be a function of \(x\)

\begin{align*} f(x) &= \sum_{n=0}^\infty x^n & \text{on the domain $|x| \lt 1$}. \end{align*}

Furthermore (also from Example 3.2.4 and Lemma 3.2.5) we know what the function is.

\begin{align*} f(x) &= \sum_{n=0}^\infty x^n = \frac{1}{1-x}. \end{align*}

Hence we can consider the series \(\sum_{n=0}^\infty x^n\) as a new way of representing the function \(\frac{1}{1-x}\) when \(|x| \lt 1\text{.}\) This series is an example of a power series.

Of course, representing a function as simple as \(\frac{1}{1-x}\) by a series doesn't seem like it is going to make life easier. However the idea of representing a function by a series turns out to be extremely helpful. Power series turn out to be very robust mathematical objects and interact very nicely with not only standard arithmetic operations, but also with differentiation and integration (see Theorem 3.5.13). This means, for example, that

\begin{align*} \diff{}{x} \left\{\frac{1}{1-x}\right\} &= \diff{}{x} \sum_{n=0}^\infty x^n & \text{provided $|x| \lt 1$} \\ &= \sum_{n=0}^\infty \diff{}{x} x^n & \text{just differentiate term by term}\\ &= \sum_{n=0}^\infty n x^{n-1}\\ \end{align*}

and in a very similar way

\begin{align*} \int \frac{1}{1-x} \dee{x} &= \int \sum_{n=0}^\infty x^n \dee{x} & \text{provided $|x| \lt 1$} \\ &= \sum_{n=0}^\infty \int x^n \dee{x} & \text{just integrate term by term}\\ &= C + \sum_{n=0}^\infty \frac{1}{n+1} x^{n+1} \end{align*}

We are hiding some mathematics under the word “just” in the above, but you can see that once we have a power series representation of a function, differentiation and integration become very straightforward.

So we should set as our goal for this section, the development of machinery to define and understand power series. This will allow us to answer questions  1 Recall that \(n!=1\times 2\times 3\times\cdots\times n\) is called “\(n\) factorial”. By convention \(0!=1\text{.}\) like

\begin{align*} \text{Is }\ e^x &=\sum\limits_{n=0}^\infty\frac{x^n}{n!} \text{ ? } \end{align*}

Our starting point (now that we have equipped ourselves with basic ideas about series), is the definition of power series.