Tests for series divergence

Ratio test for absolute convergence

The ratio test for absolute divergence of infinite series uses the limit of the magnitude of the ratio of subsequent terms in a series to bound the (absolute) series geometrically and therefore deduce absolute convergence. Using the ratio

it can be shown1 that for

  1. the series is absolutely convergent.
  2. the series is divergent2
  3. no information is given.

Note that in the case of conditional convergence, .

#state/tidy | #SemBr

Footnotes

  1. 2022. MATH1012: Mathematical theory and methods, pp. 129ff ↩

  2. This already follows from the Test for divergence by sequence limit. ↩