### Infinite series: Final chapter

### Overview common series and convergence tests

We finish with an overview of all common series and convergence tests from this chapter.

We discussed the following common series:

Geometric seriesFor \(|r|<1\) we have \[ \sum_{n=0}^\infty r^n = \frac{1}{1-r}\text. \]

Hyperharmonic seriesIt holds that \[ \sum_{n=0}^\infty \frac{1}{n^p} <\infty \iff p > 1\text. \]

A special case is \(p = 1\):

Harmonic seriesIt holds that \[ \sum_{n=0}^\infty \frac{1}{n} = \infty \text. \]

Furthemore, we proved the following convergence tests:

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 \(\textstyle\sum b_n = \infty\).

Let \(\textstyle\sum a_n\) be a series and define \(\alpha = \limsup \sqrt[n]{|a_n|}\) .

(1) If \(\alpha < 1\) then the series converges.

(2) If \(\alpha > 1\) then the series diverges.

Ratio test Let \(\textstyle\sum a_n\) be a series.

(1) If \(\textstyle\limsup \left|\frac{a_{n+1}}{a_n}\right| < 1\) then the series converges.

(2) If \(\textstyle\liminf \left|\frac{a_{n+1}}{a_n}\right| > 1\) then the series diverges.

Integral test

If \((a_n)_{n=1}^\infty\) is a decreasing sequence of positive numbers with \(\lim_{n\to\infty} a_n = 0\) then the series \( \sum_{n=1}^\infty (-1)^n a_n\) converges.