I was reading this question on this website.
Let $\{ a_n\} $ be a sequence of non-negative real numbers such that
the series $ \sum_{n=1}^{\infty}a_n $ is convergent.If $p $ is a real number such that the series $ \sum{\frac{\sqrt
a_n}{n^p}} $ diverges, then(A) $p$ must be strictly less than $\frac{1}{2} $
(B) $p$ must be strictly less than or equal to $\frac{1}{2} $
(C) $p$ must be strictly less than or equal to 1 but can be greater
than $\frac{1}{2} $(D) $p$ must be strictly less than 1 but can be greater than or equal
to $\frac{1}{2} $.
I spent a lot of time thinking about the problem but after lot of mental struggle I gave up. I looked at the answers posted and it was a simple application of Cauchy-Schwarz inequality.
Being a bit more explicit, the Cauchy-Schwarz inequality and the
assumptions imply$$\infty = \sum_{n=1}^\infty \frac{a_n^{1/2}}{n^p} \leq \left (
\sum_{n=1}^\infty a_n \right )^{1/2} \left ( \sum_{n=1}^\infty
\frac{1}{n^{2p}} \right )^{1/2}.$$
In short my question is what are some red flags to notice which signals that Cauchy-Schwarz inequality can be used to solve the problem.
I know the theorem and proof as well.
P.S. Sorry for my English. Feel free to edit.
Best Answer
You've got an arbitrary convergence series $\sum a_n$, and you're trying to figure out the convergence of $\sum \frac{\sqrt{a_n}}{n^p}$. Two things strike me about that series.
Firstly, the terms are related to $\sqrt{a_n}$, rather than just $a_n$. That will change convergence, in particular, it will make convergence slower (or make convergence not happen at all). If we want to relate this series back to $\sum a_n$, it'd be really handy if there was some inequality that naturally involved looking at the sum of squared terms, such as the Cauchy-Schwarz inequality.
Secondly, we're taking two simple series, and multiplying them term-by-term, sort of like an infinite dot product. Working with that is hard. If there was an inequality that could separate the term-by-term product of series, we might have something to work with. The Cauchy-Schwarz inequality might help there.