i dont know know how to use the limit comparison to show whether or not integral of 1/[x^(1/3) + x^(1/2) + x)] from 1 to inifinitty
When it's hard to bound, apply the limit comparison test, but this is not hard to bound because by putting $\displaystyle x=u^6$ the integral is $\displaystyle \int_{1}^{\infty }{\frac{6u^{5}}{u^{2}+u^{3}+u^{6}}\,du}$ and for all $\displaystyle u\ge1$ it's $\displaystyle \frac{u^{5}}{u^{2}+u^{3}+u^{6}}\ge \frac{u^{5}}{3u^{6}}=\frac{1}{3u}$ thus the integral diverges by direct comparison with $\displaystyle \int_{1}^{\infty }{\frac{du}{u}}.$