Overview common series and convergence tests

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:

For $|r|<1$ 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:

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.

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.

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.