Ratio test for absolute convergence

The ratio test for absolute divergence of infinite series $\sum \infty 𝑛 = 1 𝑎 𝑛$ 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

𝐿 = l i m 𝑛 \to \infty ∣ 𝑎 𝑛 + 1 𝑎 𝑛 ∣

it can be shown¹ that for

$𝐿 < 1$ the series $\sum \infty 𝑛 = 1 𝑎 𝑛$ is absolutely convergent.
$𝐿 > 1$ the series $\sum \infty 𝑛 = 1 𝑎 𝑛$ is divergent²
$𝐿 = 1$ no information is given.

Note that in the case of conditional convergence, $𝐿 = 1$ .

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2022. MATH1012: Mathematical theory and methods, pp. 129ff ↩
This already follows from the Test for divergence by sequence limit. ↩

Ratio test for absolute convergence

Footnotes