Limit comparison test

The limit comparison test takes the limit of the ratio between corresponding terms of two sequences in order to compare the rates at which the sequences go to zero. Given two infinite series $\sum \infty 𝑛 = 1 𝑎 𝑛$ and $\sum \infty 𝑛 = 1 𝑏 𝑛$ such that $𝑎 𝑛 \geq 0$ and $𝑏 𝑛 > 0$ for sufficiently large $𝑛$ , we find the positive ratio

𝑐 = l i m 𝑛 \to \infty 𝑎 𝑛 𝑏 𝑛

which gives us three cases:¹

$0 < 𝑐 < \infty$ means $𝑎 𝑛 \approx 𝑐 \cdot 𝑏 𝑛$ for large $𝑛$ , and therefore the series approach scalar multiples of each other. Therefore, they are either both convergent or both divergent.

\infty \sum 𝑛 = 1 𝑎 𝑛 \in ℝ ⟺ \infty \sum 𝑛 = 1 𝑏 𝑛 \in ℝ

$𝑐 = 0$ means $𝑏 𝑛$ becomes much larger than $𝑎 𝑛$ in the long run, and therefore the series $\sum \infty 𝑛 = 1 𝑏 𝑛$ is an upper bound on $\sum \infty 𝑛 = 1 𝑎 𝑛$ . Hence if $\sum \infty 𝑛 = 1 𝑏 𝑛$ converges $\sum \infty 𝑛 = 1 𝑎 𝑛$ converges.

\infty \sum 𝑛 = 1 𝑎 𝑛 \in ℝ ⟸ \infty \sum 𝑛 = 1 𝑏 𝑛 \in ℝ \infty \sum 𝑛 = 1 𝑎 𝑛 \notin ℝ ⟹ \infty \sum 𝑛 = 1 𝑏 𝑛 \notin ℝ

$𝑐 = \infty$ means the opposite of the above case, and hence if $\sum \infty 𝑛 = 1 𝑎 𝑛$ converges $\sum \infty 𝑛 = 1 𝑏 𝑛$ converges.

\infty \sum 𝑛 = 1 𝑎 𝑛 \in ℝ ⟹ \infty \sum 𝑛 = 1 𝑏 𝑛 \in ℝ \infty \sum 𝑛 = 1 𝑎 𝑛 \notin ℝ ⟸ \infty \sum 𝑛 = 1 𝑏 𝑛 \notin ℝ

This is advantageous over the similar Comparison test in cases where two sequences approach scalings of each other as $𝑛 \to \infty$ .

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2023. MATH1012: Mathematical theory and methods, p. 126 ↩

Limit comparison test

Footnotes