For example, $\sum_{n=0}^\infty z^n$ is known to converge for $\lvert z\rvert <1$ and to diverge for $\lvert z\rvert >1$. You could say that since $\frac{1}{2}\leq\frac{n+1}{n+2}\leq1$, then $\frac{1}{2}\sum_{n=0}^\infty z^n\leq\sum_{n=0}^\infty \frac{n+1}{n+2}z^n\leq\sum_{n=0}^\infty z^n$. One of these inequalities shows convergence at $\lvert z\rvert <1$, and the other shows divergence at $\lvert z\rvert >1$ (it’s not hard to figure out which is which).