_Posted on May 16, 2024_ _Updated on September 20, 2024_ ##### Introduction The comparison test is a useful tool towards determining the values that various series converge to: > [!theorem] **Theorem:** Let $\sum a_n$ be a series where $a_n \geq 0$ for all $n \in \mathbb{N}$. > 1. If $\sum a_n$ converges and $|b_n| \leq a_n$ for all $n$, then $\sum b_n$ converges. > 2. If $\sum a_n = \infty$ and $b_n \geq a_n$ for all $n$, then $\sum b_n = \infty$. ##### Proofs __Proof of (1):__ Suppose the series $\sum a_n$ converges and $|b_n| \leq a_n$ for all $n$. By the [[Zeno's dichotomy paradox and a definition of convergence for series|definition of convergence for series]], the sequence of partial sums converges for all $n \in \mathbb{N}$, so that we have an $M$ such that $0 \leq \sum_{k = m}^n |b_k|\leq \sum_{k=m}^na_k \leq M$by the [[Convergent Sequences are bounded|boundedness of convergent sequences]]. Thus the sequence of partial sums $(\sum_{k=m}^n|b_k|)$ is bounded below by 0 and above by $M$. Additionally, since $|b_n| \geq 0$ for all $n$, the sequence of partial sums $(\sum_{k=m}^n|b_k|)$ is monotone increasing [[Bounded Monotone Sequences Converge, and Absolutely Convergent Series Converge|and therefore convergent]]. Hence, the series $\sum |b_k|$ converges. Therefore, the series $\sum b_k$ converges by [[Bounded Monotone Sequences Converge, and Absolutely Convergent Series Converge|absolute convergence]] $\blacksquare$ ---- Alternate proof (using cauchy criterion): (I will eventually add this ( #TODO )) --- __Proof of (2):__ Suppose $\sum a_n = \infty$, and $b_n \geq a_n$ for all $n$. Since $b_n \geq a_n$ for all $n$, $\sum_{k=m}^nb_k \geq \sum_{k=m}^na_k$ for all $n$. By the definition of convergence for series, the sequence of partial sums $(\sum_{k=m}^na_k)$ diverges to $\infty$. Thus, for every real number $M > 0$, there exists $N \in \mathbb{N}$ such that for all $n > N$, $\sum_{k=m}^nb_k \geq \sum_{k=m}^n a_k > M$. Therefore, $\sum_{k=m}^n b_k > M$. We conclude that $\sum b_n = \infty$. $\blacksquare$