**Lemma**: Suppose that $0\leq a_{n} \leq b_{n}$ for every $n$. Then: - if $\sum b_{n} < \infty$ then $\sum a_{n} < \infty$ , - if $\sum a_{n} = \infty$ then $\sum b_{n} = \infty$ . **Proof**: 1. Write $\alpha_{k} = \sum_{n=1}^{k} a_{n}$ and $\beta_{k} = \sum_{n=1}^{k} b_{n}$. Since $a_{n} \leq b_{n}$ for every $n$, $\alpha_{k} \leq \beta_{k}$ for every k If $\sum b_{n} < \infty$ , then $\alpha_{k} \leq \beta_{k} \leq \sum b_{n}$ for every $k$, So $(\alpha_{k})$ is an increasing sequence that is bounded above and so converges by [[Monotone Bounded Real Sequence is Convergent|monotone convergence theorem]]. 2. Conversely, if $\sum a_{n}$ does not converge then the partial sums $\alpha_{k}$ are not bounded above, so $(\beta_{k})$ is not bounded above So the series $\sum \beta_{n}$ must diverge. We can improve this [[Comparison Test for Series With Non-negative Terms (Corollary 1)]]. # Applications **Corollaries**: By [[Comparison Test for Series With Non-negative Terms (Corollary 1)]], ... By [[Comparison Test for Series using Absolute Value of Terms (Corollary 2)]], ....