Comparison Test

This tests utilises the ‘bounds’ of other series, usually geometric or p-harmonic, to test a given series. It’s similar to the Sandwich Theorem, but doesn’t require two other series, since a series can either converge to a value or diverge (approach infinity).

Definition §

Let $a_n$ be a sequence of real numbers with positive terms only, and ${\sum }_{n=1}^{\infty }{a}_n$ be it’s respective series.

1. If there exists another series, ${\sum }_{n=1}^{\infty }{b}_n$, such that $0\le b_n\le a_n\forall n$ then if ${\sum }_{n=1}^{\infty }{b}_n$ diverges, ${\sum }_{n=1}^{\infty }{a}_n$ also diverges.

2. Similarly, if there exists a series, ${\sum }_{n=1}^{\infty }{c}_n$ such that $c_n\ge a_n\forall n$ then if ${\sum }_{n=1}^{\infty }{c}_n$ converges, ${\sum }_{n=1}^{\infty }{a}_n$ also converges

It’s best to use geometric or harmonic series because their conditions for converging or diverging are simple.

https://www.desmos.com/calculator/8op6xpfuks

Examples §

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