Is series convergent/divergent

Question

I need to find out is series convergent or not $$ \sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}} $$ How can I do that? Can you show step-by step solution?

Answer 1

The terms are positive, so the strategy "find an equivalent" works. We can show that $$\frac{5k-2}{(3k^2-2)\sqrt[3]{k+6}}\sim\frac{5k}{3k^2k^{1/3}}=\frac 53\frac{1}{k^{4/3}}.$$

Answer 2

HINTS:

• $5k - 2 < 5k$ for all $k$
• $3k^2 - 2 > 2k^2$ for $k > 1$
• $\sqrt[3]{k + 6} > \sqrt[3]{k}$ for all $k$

Answer 3

$$\frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}}=\Theta\left(\frac{k}{k^2\cdot\sqrt[3]{k}}\right)=\Theta\left(\frac1{k^{4/3}}\right)\qquad \&\qquad 4/3\gt1$$ Sorry, no other step...