### Infinite series: Convergence tests

### The comparison test

We can estimate series just like we can estimate normal limits. The following two propositions are together called the comparison test.

The comparison test (1) Let\(\textstyle\sum a_n\) and \(\textstyle\sum b_n\) be series such that \(a_n\geq 0\) for all \(n\). If \(\textstyle\sum a_n\) converges and \(|b_n|\leq a_n\) for all \(n\) then \(\textstyle\sum b_n\) converges as well.

The comparison test (2) Let \(\textstyle\sum a_n\) and \(\textstyle\sum b_n\) be series. If \(\textstyle\sum a_n = \infty\) and \(b_n\geq a_n\) for all \(n\) then \(\textstyle\sum b_n = \infty\).

The first part of the comparison test can be used to prove that a series converges, while the second part can be used to show that a series diverges.

The series \(\sum_{n=1}^\infty \frac{1}{2n+1}\) diverges. This is because \[ \frac{1}{2n+1} > \frac{1}{2n+n} = \frac{1}{3n} \] and \(\textstyle\sum \frac{1}{3n} = \frac{1}{3} \sum \frac{1}{n}\) diverges.