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.
