# Comparison Test

## Theorem

Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers.

Let $\sequence {a_n}$ be a sequence $\R$ or sequence in $\C$.

Let $\forall n \in \N_{>0}: \cmod {a_n} \le b_n$.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.

### Corollary

Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers.

Let $\left \langle {a_n} \right \rangle$ be a sequence in sequence $\R$ or sequence in $\C$.

Let $H \in \R$.

Let $\exists M: \forall n > M: \left\vert {a_n}\right\vert \le H b_n$.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges.

## Proof

Let $\epsilon > 0$.

As $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero.

So:

$\displaystyle \exists N: \forall n > N: \sum_{k \mathop = n + 1}^\infty b_k < \epsilon$

Let $\sequence {a_n}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Then $\forall n > m > N$:

 $\displaystyle \cmod {a_n - a_m}$ $=$ $\displaystyle \cmod {\paren {a_1 + a_2 + \cdots + a_n} - \paren {a_1 + a_2 + \cdots + a_m} }$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \cmod {a_{m + 1} + a_{m + 2} + \cdots + a_n}$ $\quad$ Indexed Summation over Adjacent Intervals $\quad$ $\displaystyle$ $\le$ $\displaystyle \cmod {a_{m + 1} } + \cmod {a_{m + 2} } + \cdots + \cmod {a_n}$ $\quad$ Triangle Inequality for Indexed Summations $\quad$ $\displaystyle$ $\le$ $\displaystyle b_{m + 1} + b_{m + 2} + \cdots + b_n$ $\quad$ $\quad$ $\displaystyle$ $\le$ $\displaystyle \sum_{k \mathop = n + 1}^\infty b_k$ $\quad$ $\quad$ $\displaystyle$ $<$ $\displaystyle \epsilon$ $\quad$ $\quad$

So $\sequence {a_n}$ is a Cauchy sequence.

The result follows from Real Number Line is Complete Metric Space or Complex Plane is Complete Metric Space.

$\blacksquare$