Provide an appropriate response.
-$\text { If } \sum a _ { n } \text { converges, what can be said about } \sum a _ { n } \mathrm { k } \text {, where } \mathrm { k } \text { is an integer greater than } 1 \text { ? }$
A) The series always converges.
B) The series may converge or diverge.
C) The series always diverges.

Question

Quizplus · Accepted Answer

It is not always true that the series $\sum a_n k$ converges. For example, consider $a_n = \frac{1}{n}$ and $k=2$, then $\sum a_n$ converges, but $\sum a_n k = \sum \frac{2}{n}$ diverges. However, if $a_n$ is a convergent series and $k$ is a positive integer, then $\sum a_n k$ also converges by the limit comparison test.

Provide an Appropriate Response A) the Series Always Converges