Alternating Series Test

Calculus 2 · Axiom Academy

LESSON The Alternating Series Test When the terms of a series flip sign and shrink to zero, the partial sums zig-zag inward and trap a limit — and the leftover error is at most one term. 1. What Is an Alternating Series? An alternating series is one whose terms flip sign every step. Writing the size of each term as a positive number b_n , the general form is: the factor (-1)^ n+1 (or (-1)^n ) supplies the alternating signs Watch the actual terms of the alternating harmonic series land one at a time: each is the opposite sign of the last, and each is smaller than the last. 2. The Test: Two Conditions (Leibniz) Decreasing: for all large n — the term sizes never grow. Limit zero: — the terms shrink all the way to nothing. The animation checks both conditions for : on the left the term sizes step down (condition 1), on the right they dive to the zero line (condition 2). 3. Why It Works: The Partial Sums Bracket the Limit Add the terms one at a time and plot each running total S_n . Because each new term is smaller than the last and flips the direction, the partial sums overshoot, then undershoot by less, then overshoot by less — closing in on the limit from both sides. The bracketing has a powerful payoff. Because the true sum S always sits between S_n and the next partial sum, the distance from S_n to S can be no bigger than the very next term you left out:

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