Comparison Test (Series)

Comparison Test

Suppose $\forall n \in N$ we have $a_{n}, b_{n} \in R$ satisfy $0 \leq a_{n} \leq b_{n}$ . Then:

Proof
Suppose $\sum b_{n}$ converges. Let's show $\sum a_{n}$ satisfies the Cauchy Condition for Series, and thus converges.

Let $ε > 0$ . Since $\sum b_{n}$ converges then we can choose $N \in N$ such that $n > m \geq N$ has it that:

| b_{m + 1} + \dots + b_{n} | < ε

Everything here is non-negative, so we can get rid of the absolute value signs:

b_{m + 1} + \dots + b_{n} < ε

Suppose this is true, that we choose our $n > m \geq N$ . Then:

\begin{aligned} | a_{m + 1} + \dots + a_{n} | & = a_{m + 1} + \dots + a_{n} \\ \leq b_{m + 1} + \dots + b_{n} \\ < ε \end{aligned}

Thus $\sum a_{n}$ satisfies the Cauchy Condition for Series, so then it must converge.
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