Answer
Example: ${a_n} = \cos \pi n$ for $n = 1,2,3,...$.
The sequence $\left\{ {{a_n}} \right\}$ diverges.
However, $\mathop {\lim }\limits_{n \to \infty } \left| {{a_n}} \right| = 1$.
So, the sequence $\left\{ {\left| {{a_n}} \right|} \right\}$ converges.
Work Step by Step
As an example, consider ${a_n} = \cos \pi n$ for $n = 1,2,3,....$.
We may write out this sequence as $-1,1,-1,1,...$.
Since the terms bounce back and forth with values $-1$ and $1$ but never settle down to approach a limit, $\left\{ {{a_n}} \right\}$ diverges.
However, $\left| {{a_n}} \right| = \left| {\cos \pi n} \right| = 1$ for $n=1,2,3,...$.
So, $\mathop {\lim }\limits_{n \to \infty } \left| {{a_n}} \right| = 1$. Thus, the sequence $\left\{ {\left| {{a_n}} \right|} \right\}$ converges.
