Tests for series divergence

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 ∑∞𝑛=1𝑎𝑛 and ∑∞𝑛=1𝑏𝑛 such that 𝑎𝑛 ≥0 and 𝑏𝑛 >0 for sufficiently large 𝑛, we find the positive ratio

𝑐=lim𝑛→∞𝑎𝑛𝑏𝑛

which gives us three cases:¹

1. 0 <𝑐 <∞ means 𝑎𝑛 ≈𝑐 ⋅𝑏𝑛 for large 𝑛, and therefore the series approach scalar multiples of each other. Therefore, they are either both convergent or both divergent.

∞∑𝑛=1𝑎𝑛∈ℝ⟺∞∑𝑛=1𝑏𝑛∈ℝ

2. 𝑐 =0 means 𝑏𝑛 becomes much larger than 𝑎𝑛 in the long run, and therefore the series ∑∞𝑛=1𝑏𝑛 is an upper bound on ∑∞𝑛=1𝑎𝑛. Hence if ∑∞𝑛=1𝑏𝑛 converges ∑∞𝑛=1𝑎𝑛 converges.

∞∑𝑛=1𝑎𝑛∈ℝ⟸∞∑𝑛=1𝑏𝑛∈ℝ∞∑𝑛=1𝑎𝑛∉ℝ⟹∞∑𝑛=1𝑏𝑛∉ℝ

3. 𝑐 =∞ means the opposite of the above case, and hence if ∑∞𝑛=1𝑎𝑛 converges ∑∞𝑛=1𝑏𝑛 converges.

∞∑𝑛=1𝑎𝑛∈ℝ⟹∞∑𝑛=1𝑏𝑛∈ℝ∞∑𝑛=1𝑎𝑛∉ℝ⟸∞∑𝑛=1𝑏𝑛∉ℝ

This is advantageous over the similar Comparison test in cases where two sequences approach scalings of each other as 𝑛 →∞.

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Footnotes

1. 2023. MATH1012: Mathematical theory and methods, p. 126 ↩