I'm trying to prove that the best moment tail bound is no worse than the best Chernoff bound. I have all pieces other than this little frustration here:

If $c\le \frac{{a}_{i}}{{b}_{i}}\mathrm{\forall}i$, then $c\le \frac{\sum _{i=1}^{\mathrm{\infty}}{a}_{i}}{\sum _{i=1}^{\mathrm{\infty}}{b}_{i}}$

If $c\le \frac{{a}_{i}}{{b}_{i}}\mathrm{\forall}i$, then $c\le \frac{\sum _{i=1}^{\mathrm{\infty}}{a}_{i}}{\sum _{i=1}^{\mathrm{\infty}}{b}_{i}}$