Home page for accesible maths Math 101 Chapter 1: Sequences and Series 1.28 Building sequences 1.30 Examples of convergent and divergent sequences

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1.29 Convergence of sequences

We often write $(a_{n})\,$ for a sequence and give a formula for the $n^{th}\,$ term. By convention, we take $n\,$ to be an integer variable; usually we start with $n=1\,$ , so the sequence is $(a_{1},a_{2},a_{3},\dots)$ .

Example

$a_{n}=n/(n+1)\,$ gives the sequence $1/2,2/3,3/4,\dots.$

Convergence of sequences. A sequence $(a_{n})\,$ of real numbers is said to converge to a real number $L\,$ as $n\rightarrow\infty\,$ , if the $a_{n}\,$ are arbitrarily close to $L\,$ for all sufficiently large $n\,$ . We write $a_{n}\rightarrow L\,$ as $n\rightarrow\infty\,$ ; and we call $L\,$ the limit of the sequence.

This definition will be discussed in detail in MATH113, so here we only look at simple examples to illustrate what it means and how it works.