Comparison and Limit Comparison

Real Analysis · Axiom Academy

If a positive series is trapped under a series you already understand, it inherits its fate — and a settling ratio lets you compare without the inequality. 1. Direct Comparison: Trapping a Series Suppose both series have positive terms. The whole idea is a squeeze : bound your unknown series against a known one , term by term, and the bound carries the verdict. Trapped below something finite finite Sitting above something infinite infinite Take . We don't know its sum offhand — but adding 1 to the denominator only makes each term smaller than the matching term of the p -series , which we know converges. 3. Limit Comparison: Letting the Ratio Decide Sometimes the inequality is fiddly. Instead, watch the ratio a_n/b_n . If it settles to a finite, positive number L , the two series are eventually proportional — so they must share the same fate. then and both converge or both diverge . Test against the harmonic series . No clean inequality is needed — just the ratio: Needs a term-by-term inequality you can actually prove. Best when the bound is obvious. Only needs . Best when terms share asymptotic behavior but the inequality is messy. You've seen how a positive series inherits the fate of a known one it's pinned against — by a trapping inequality, or by a ratio that settles. Scroll up to revisit any step.

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