# Determine whether the series $\sum_{n =1}^{\infty} \frac{n + \sqrt{n}}{2n^3 -1}$ converges?

Determine whether the series $$\sum_{n =1}^{\infty} \frac{n + \sqrt{n}}{2n^3 -1}$$ converges?

My answer: For large n, the given series is smaller than $$\sum_{n =1}^{\infty}\frac {2n}{2n^3}$$ which is equal to $$\sum_{n =1}^{\infty}\frac {1}{n^2}$$, but we know that the later is convergent by the p-series test, then by comparison test the given series is convergent.

• It is correct, however you glossed over the exact details of how we know $\frac{n+\sqrt{n}}{2n^3-1}$ is smaller than $\frac{2n}{2n^3}$ for large enough $n$. It would be best to show this more explicitly. – JMoravitz Jan 15 at 5:39
• It is correct. In fact the inequality holds for all $n$. Just verify it carefully to make the argument complete. – Kabo Murphy Jan 15 at 5:39
• @KaviRamaMurthy I think it does not hold for $n=1$ – Secretly Jan 15 at 5:42
• @hopefully Right. It is true for all $n>1$. – Kabo Murphy Jan 15 at 5:45
• Also, because the Comparison Test applies only to series with nonnegative terms, you should mention that $\frac{n + \sqrt{n}}{2n^3 -1}>0$. – Taladris Jan 15 at 5:49

You have to make the denominator smaller, not larger. So yse $$n^3$$ and everything is fine.