Determine whether the following statement is true or false. If the statement is true, prove it$,$ and, if it is false, provide a counterexample.

If ${\displaystyle \underset{\text{n}\text{i}\text{n}\text{N}}{\overset{?}{}}}{a}_n$ is conditionally convergent, then ${\displaystyle \underset{\text{n}\text{i}\text{n}\text{N}}{\overset{?}{}}}{n}^k{a}_n$ is divergent, for each kin$(1,+).$