Comparison Tests

Calculus 2 · Axiom Academy

Decide whether a series converges by sizing it up against one whose fate you already know. Suppose two series have nonnegative terms and yours, a_n , is pinned underneath a known series b_n : that is for all n past some point N . The known ceiling controls the floor. If converges , then converges. Intuition. Think of fitting boxes in a trunk. If a heavier box fits, the lighter one surely fits too. If the lighter box won't fit, the heavier one has no chance. The bigger series caps the smaller one from above; 0 caps it from below. 2. Direct Comparison in Action Solution. Adding 5 to the denominator only makes each fraction smaller, so for every : The benchmark is a convergent p -series ( ). Our terms sit strictly below it, so by Direct Comparison the smaller series must converge too. Watch each blue a_n bar stay shorter than its green b_n = 1/n^2 partner, while the running total of a_n stays trapped under the known ceiling. Direct comparison needs a clean inequality, and sometimes that inequality points the wrong way or is a pain to prove. The Limit Comparison Test trades the inequality for a single limit of the ratio of terms. Intuition. If the ratio of terms settles onto a positive constant, the two series are "proportional" for large n — one is roughly L times the other, term for term — so they share the same long-run fate. Solution. For large n the numerator behaves like 3n^2 and the denominator like 5n^4 , so . That points us at the benchmark .

This is the written version of the interactive lesson above. See the full Calculus 2 course.