Suppose we have a positive number sequence $(a_n)_{n≥1}$, which produces a converging series $\sum_{n=1}^∞ a_n$.
How do I prove a series $\sum_{n=1}^∞ \frac {a_n}{\sqrt{n}}$ converges as well?
I tried the ratio test, but I cant get anywhere with it.
I've also tried using limits, but these prove nothing.
Lastly, I've tried the root test, getting to this $$\sum_{n=1}^∞ \frac {a_n}{\sqrt{n}}≤ \sqrt{\sum_{n=1}^∞ a_n^2} \sqrt{\sum_{n=1}^∞ \frac {1}{n}}< ∞,$$ but I don't know what to assume and how to proceed with it.
I'm certain that I'm doing something wrong and the way to solve it is with the previous tests, but I'm struggling to see my mistakes.
Can anyone provide some insight?
Is it a sequence with positive terms? If that is the case, since you have that $0 \leq \frac{a_n}{\sqrt{n}}\leq a_n$ and $\sum a_n$ converges, $\sum \frac{a_n}{\sqrt{n}}$ also converges.