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Section 1.2 Calculus

The first of the basic ideas we're going to need is improper integration, which is a fancy way of saying an integral with infinity as a limit. Do not be fooled by the notation

$\begin{equation*} \int_a^\infty f(x) \, dx. \end{equation*}$

Infinity is not a number, and the notation for improper integrals is merely shorthand for a limit (this is a recurring theme with infinity). That is, we say that an improper integral in infinity converges if

$\begin{equation*} \int_a^\infty f(x) \, dx = \lim_{N \to \infty} \int_a^N f(x) \, dx \text{ exists and is finite}. \end{equation*}$

DO NOT treat infinity like a number.

An infinite series is also a limit. (The clue is that infinity is present in the symbol). That is,

$\begin{equation*} \sum_{n=0}^\infty a_n = \lim_{N \to \infty} \sum_{n=a}^N a_n. \end{equation*}$

This is sometimes called the “limit of partial sums”. An infinite series exists if the limit of partial sums exists and is finite. An infinite series

$\sum a_n$ is said to converge absolutely if

$\sum \abs{a_n}$ is a convergent series.

You learned many ways of determining when an infinite series converges in calculus. The most important test for us in this course is the ratio test, which states that a series converges (absolutely) if

$\begin{equation*} \lim_{n \to \infty} \abs{\frac{a_{n+1}}{a_n}} \lt \infty. \end{equation*}$