If $0\le a_n\le b_n$, convergence of $\sum b_n$ implies convergence of $\sum a_n$, and divergence of $\sum a_n$ implies divergence of $\sum b_n$.Suggest a slogan
Let \((a_n)_{n\geq1}\) and \((b_n)_{n\geq1}\) be real sequences with \(0\leq a_n \leq b_n\) for all \(n\).
If \(\sum_n b_n < \infty\), then \(\sum_n a_n < \infty\).
If \(\sum_n a_n = \infty\), then \(\sum_n b_n = \infty\).