what would be an appropriate test (out of ratio, intergral, limit comparison) to decide if the series
sum of(1 to infinity)[k^(1/2)/$\displaystyle (3*k^3+1)$]
converges or diverges?
you can use ratio test I prefer it
$\displaystyle lim_{k\rightarrow\infty}\frac{\sqrt{k+1}(3(k)^3)}{ ( \sqrt{k}(3(k+1)^3+1)}=lim_{k\rightarrow\infty}\fra c{\sqrt{k+1}(3k^3)}{( \sqrt{k}(3k^3+3k^2+3k+1+1)}$
$\displaystyle =lim_{k\rightarrow\infty}\frac{\frac{\sqrt{k+1}}{k ^3}(3)}{\frac{\sqrt{k}}{k^3}(3+\frac{3}{k})+\frac{ 3}{k^2}+\frac{2}{k^3})}$
$\displaystyle =lim_{k\rightarrow\infty}\frac{\frac{\sqrt{k+1}}{k ^3}(3)}{\frac{\sqrt{k}}{k^3}(3+\frac{3}{k})+\frac{ 3}{k^2}+\frac{2}{k^3})}=1$
$\displaystyle the.... test... failed ...try.. comparison... test$