Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.07 Relationship to Sequences of Absolute Values

The following theorem can be used to evaluate limits of sequences when the sign of the terms in the sequence alternate between positive and negative.

Theorem: Relationship to Sequences of Absolute Values
The limit $\underset{n\to \mathrm{\infty }}{\lim }{a}_n$ equals zero if and only if $\begin{array}{r}\underset{n\to \mathrm{\infty }}{\lim }|a_n|=0.\end{array}$

In other words, if we are given the sequence

$\{a_n{\}}_{n=1}^{\mathrm{\infty }}$

and

$\underset{n\to \mathrm{\infty }}{\lim }|a_n|=0$

then

$\begin{array}{r}\underset{n\to \mathrm{\infty }}{\lim }{a}_n=0\end{array}$

The converse is also true but we will not use it. For us, this theorem is only useful as a test for convergence.