I tried to prove the following:
Suppose that $a_n$ is a sequence of non-negative numbers and that the series $\sum a_n$ converges. Prove that the series $\sum a_n^p$ converges for p>1.
How should I start this prove? Should I break down $a_n$ into three cases: $a_n$ equals 1, less then 1, greater than 1? or Should I use comparison test?
Best Answer
Well, you already gave the answer in your question.
If $\sum a_n$ converges, $\{a_n\}$ converges to zero. As we’re dealing with non negative series and $p \gt 1$, we have for all $n $ large enough
$$0\le a_n^p \le a_n$$ and therefore $\sum a_n^p$ converges.