Proposition: Limit Comparizon Test

Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be infinite series with positive terms $a_n > 0$ and $b_n > 0$ for all $n\in\mathbb N$ and let the sequence of ratios $\left(\frac{a_n}{b_n}\right)_{n\in\mathbb N}$ converge to a positive limit $$\lim_{n\to\infty }\frac{a_n}{b_n}=c > 0.$$ Then the two series are either both convergent or both divergent. If the limit is zero ($c=0$), then the convergence of $\sum_{n=0}^\infty b_n$ implies the convergence of $\sum_{n=0}^\infty a_n.$

Proofs: 1

Proofs: 1

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition