Relationship to Sequences of Absolute Values

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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 $\text{[math]}$ equals zero if and only if $\text{[math]}$

Theorem: Relationship to Sequences of Absolute Values

The limit

$\text{[math]}$

equals zero if and only if

$\text{[math]}$

In other words, if we are given the sequence

$\text{[math]}$

and

$\text{[math]}$

then

$\text{[math]}$

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

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