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With nth term test $ \sum a_n $ diverges but what about $ \sum \frac{a_n}{n}$ can we use comparison test by taking auxillary series $\sum \frac{1}{n}$?

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Yes you're right. By the limit test the series $\sum a_n$ is divergent since $\lim\limits_{n\to\infty}a_n\ne0$ and using the asymptotic comparison we have $$\frac{a_n}n\sim_\infty \frac an$$

and the harmonic series $\sum\frac1n$ so the given series is also divergent.

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Yes you can. For some large enough $N$, for $n > N$, we have $a_n > \frac{a}{2}$. Then

$$\sum_{n>N} \frac{a_n}{n} \geq \frac{a}{2} \sum_{n>N} \frac{1}{n} \to \infty$$

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$$\sum_{n=1}^\infty {1\over n}$$ diverges so if $a\neq 0$ it diverges.

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