Tests for series divergence

The following is a summary of tests available for the divergence of infinite series.

Name	Principle	When to use
Test for divergence by sequence limit	$l i m 𝑛 \to \infty 𝑎 𝑛 \neq 0 ⟹$ DNE	Obvious cases
Integral test	$\sum \infty 𝑛 = 1 𝑎 𝑛$ and $\int \infty 1 𝑓 (𝑥) 𝑑 𝑥$ both converge or diverge	$𝑎 𝑛$ is extendible to an integratable positive decreasing function
Comparison test	Analogous to the squeeze theorem	A divergent lower bound or convergent upper bound is known.
Limit comparison test	Compare the ratio between two sequences	Same as above, but series which approach multiples of each other are especially useful
Alternating series test	Test for divergence becomes necessary and sufficient for alternating series	Any alternating sequence, which is absolutely non-increasing for large $𝑛$
Absolute convergence	Absolutely convergent $⟹$ convergent	$\sum \infty 𝑛 = 1 \| 𝑎 𝑛 \|$ is convergent
Ratio test for absolute convergence	A ratio of subsequent terms less than one implies the series is bound by the geometric series	Absolute ratio of subsequent terms can be found.

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