How to prove $\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$ diverges?

Question

$$\sum_{n=1}^{\infty} \frac{3}{\sqrt[3]{n^2+2}}$$

It seems clear to me that this series diverges because the dominant temr is $1/n^{2/3}$, a p-series with $p < 1$

However I need to prove divergence using something like the comparison test, integral test, or similar.

I can't work out a suitable comparison to make to prove divergence. Suggestions?

Answer 1

Note that for all positive integers $n$, $$n^2 + 2 < n^2 + 4n + 4 = (n+2)^2.$$ Therefore, $$\frac{3}{\sqrt[3]{n^2 + 2}} > \frac{3}{\sqrt[3]{(n+2)^2}} > \frac{1}{(n+2)^{2/3}},$$ hence $$\sum_{n=1}^\infty \frac{3}{\sqrt[3]{n^2+2}} > \sum_{n=1}^\infty \frac{1}{(n+2)^{2/3}} = - 1^{2/3} - \frac{1}{2^{2/}} + \sum_{n=1}^\infty \frac{1}{n^{2/3}},$$ the last sum of which is a divergent $p$-series, hence the given sum also diverges.

Answer 2

You have $${3\over \root{3}\of{n^2 + 2}}\sim {3\over{n^{2/3}}}.$$ Now apply the limit comparsion and $p$-series theorems.