## The Comparison Test

Theorem 6.4.1 (The Comparison Test)   Suppose and are series with all and positive and for each .
1. If converges, then so does .
2. If diverges, then so does .

Proof. [Proof Sketch] The condition of the theorem implies that for any ,

from which each claim follows.

Example 6.4.2   Consider the series . For each we have

Since converges, Theorem 6.4.1 implies that also converges.

Example 6.4.3   Consider the series . It diverges since for each we have

and diverges.

William Stein 2006-03-15