Comparison tests

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

For positive terms: if $a_k \le b_k$ and $\sum b_k$ converges, so does $\sum a_k$ . The limit version: if $a_k / b_k \to c \in (0, \infty)$ , both series behave the same. In practice, compare with geometric or $p$ -series.

Formulas

0 \le a_k \le b_k,\ \sum b_k < \infty \implies \sum a_k < \infty

\frac{a_k}{b_k} \to c \in (0,\infty) \implies \Big(\sum a_k < \infty \iff \sum b_k < \infty\Big)

Where it shows up in computing

Algorithm analysis and complexity★★★★★frequentScientific computing and algorithms
Bounding a messy cost sum by a simpler one is exactly how asymptotic bounds are proved.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Algorithm analysis and complexity★★★★★
Absolute and conditional convergence→Floating point (IEEE 754)★★★★★
Absolute and conditional convergence→Floating point (IEEE 754)→Scientific computing★★★★★

What depends on it

Absolute and conditional convergence

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.