DCT/LCT Rules

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9 Terms

1
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aₙ ≤ bₙ and Σbₙ converges

Σaₙ converges

2
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aₙ ≥ bₙ and Σbₙ diverges

Σaₙ diverges

3
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aₙ ≤ bₙ with Σbₙ diverging

nothing

4
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aₙ ≥ bₙ with Σbₙ converging

nothing

5
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For LCT to work, terms eventually must be:

positive

6
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p-series Σ1/nᵖ:

converges if p > 1, diverges if p ≤ 1

7
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Geometric Σrⁿ

converges if |r| < 1, diverges if |r| ≥ 1

8
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For LCT to be used:

aₙ, bₙ > 0

9
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If the limit from LCT falls in these bounds, Σaₙ and Σbₙ do the same thing:

0 < L < ∞