To determine whether $\sum_{n=1}^\infty \frac{a_n}{n}$ is convergent, let's analyze the conditions:

1. Given:

   • $a_n \geq 0$, i.e., $a_n$ is non-negative.
   • $\sum_{n=1}^\infty a_n^2$ is convergent.

   This means that the series of squares of the $a_n$ terms is convergent.

2. To Test:

   • Is $\sum_{n=1}^\infty \frac{a_n}{n}$ convergent?

Key Concepts:

• Since $a_n^2$ is convergent, it follows by the Cauchy-Schwarz inequality (or other relevant tests) that $a_n$ behaves sufficiently small as $n \to \infty$, particularly that it decays at a certain rate.
• A comparison with $\frac{1}{n}$-terms arises naturally.

Applying the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality for two sequences $\{a_n\}$ and $\{b_n\}$ states:
$\left( \sum_{n=1}^\infty a_n b_n \right)^2 \leq \left( \sum_{n=1}^\infty a_n^2 \right) \left( \sum_{n=1}^\infty b_n^2 \right).$ Here:

• Let $a_n = a_n$ (as given).
• Choose $b_n = \frac{1}{n}$.

Thus: $\left( \sum_{n=1}^\infty \frac{a_n}{n} \right)^2 \leq \left( \sum_{n=1}^\infty a_n^2 \right) \left( \sum_{n=1}^\infty \frac{1}{n^2} \right).$

Justification:

1. The series $\sum_{n=1}^\infty a_n^2$ converges (given).
2. The series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges, as it is a p-series with $p = 2 > 1$.

Since the right-hand side is finite, it follows that: $\sum_{n=1}^\infty \frac{a_n}{n}$ is finite, i.e., the series converges.

Final Answer:

The series $\sum_{n=1}^\infty \frac{a_n}{n}$ is convergent.
Correct option: (a) convergent.

Do you need further clarification on the steps? Let me know if you'd like more details.

Related Questions:

1. What is the Cauchy-Schwarz inequality and how is it applied in series problems?
2. How does a p-series test determine whether a series converges or diverges?
3. Why does $\sum a_n^2$ converging impose restrictions on $a_n$'s behavior?
4. Can $\sum \frac{a_n}{n}$ still converge if $a_n^2$ diverges? Why or why not?
5. How can other inequalities (like Hölder's inequality) be used to determine series convergence?

Tip:

When analyzing convergence of series involving powers or fractions, always consider comparison tests, inequalities, and behavior of related p-series.