The Limit Comparison Test

When two series have terms that behave the same way for large n, they should converge or diverge together, even if one is always a little bigger or smaller than the other term by term. The direct comparison test needs a clean inequality for every n, which is often messy to prove or just false near the start. Limit comparison sidesteps that: instead of comparing term sizes directly, you compare their long-run behavior by taking the limit of a_n over b_n as n goes to infinity.

If lim (a_n / b_n) = L, where L is a positive finite number, the two series are tied together — a_n behaves like a constant multiple of b_n for large n, so whatever b_n's series does, a_n's series does too. You pick b_n to be a simpler series, usually a p-series like 1/n or 1/n², whose convergence you already know. Find the limit of the ratio, check that it is positive and finite, then read off the same conclusion for your original series. If the limit is 0 or infinite, the test gives no conclusion, and you need another method.

This test matters most when the series looks almost like a p-series but has extra terms cluttering the numerator or denominator, the kind of thing that makes direct comparison inequalities painful. It sits alongside the direct comparison test and the integral test as tools for series with positive terms, and it leans on your fluency with p-series and the harmonic series as the known benchmarks everything else gets measured against.