Knowlpedia

Limit Comparison Test

Two positive series with asymptotically proportional terms converge or diverge together.

Limit comparison test: Let ∑n=1∞an\sum_{n=1}^\infty a_n and ∑n=1∞bn\sum_{n=1}^\infty b_n be with an>0a_n>0 and bn>0b_n>0 for all sufficiently large nn. Suppose the limit

lim⁡n→∞anbn=c\lim_{n\to\infty}\frac{a_n}{b_n}=c

exists and satisfies 0<c<∞0<c<\infty. Then ∑an\sum a_n converges if and only if ∑bn\sum b_n converges.

Remarks

This is often used when the is too rigid, but one can identify the asymptotic size of ana_n relative to a known bnb_n (such as a pp-series or a geometric series).