Comparison Test

Theorem
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 $\R$ or sequence in $\C$.

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

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

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 $\left \langle s_n \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Then $\forall n > m > N$:

So $\left \langle s_n \right \rangle$ is a Cauchy sequence.

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

Also see

 * Comparison Test for Divergence