Sequences and Series

Abstract

A sequence is a function whose domain is a set of the form $\{n\in \mathbb {Z}\ : \ n\ge m\}$; m is usually 1 or 0. Thus, a sequence is a function that has a specified value for each integer $n\ge m$. It is customary to denote a sequence by a latter such as a and to denote its value at n as $a_n$ rather than a(n). It is often convenient to write the sequence as $\{a_n\}_{n=m}^\infty$. If $m=1$ we may write $\{a_n\}_{n\in \mathbb {N}}$. We study the sequences whose range values are real numbers; i.e., each $a_n$ represent a real number. Sometimes we write $\{a_n\}$ when the domain is understood. A sequence $\{a_n\}$ of real numbers is said to be convergent to a real number L provided that: For every $\epsilon >0$ there exists a number N such that $n>N$ implies $|a_n-L|<\epsilon$.