Alternating Series Test

Real Analysis · Axiom Academy

LESSON The Alternating Series Test When the signs alternate and the terms shrink to zero, the partial sums oscillate into the limit — bracketing it in nested shrinking intervals. Take an alternating series with every b_n > 0 . Leibniz s test asks just two things of the magnitudes b_n : If both hold, the series converges . Watch why with the alternating harmonic series, where b_n = 1/n : each partial sum lands on the opposite side of the limit, and by a shorter step than the one before — a staircase squeezing onto . Here is the engine behind the test. Each consecutive pair of partial sums S_n and S_ n+1 traps the true sum S between them, and the next sum always lands inside the previous bracket. So the brackets are nested: The even sums rise; the odd sums fall; and the gap between them is exactly the term . Two monotone sequences squeezing shut on a single point: that point is S . The nesting hands us a free, practical bonus. Stop at the partial sum S_n ; the true sum S lies in the very next bracket, on the far side of S_n , no further than one step away. That step is the first omitted term, b_ n+1 . Hence the truncation error bound : You ve seen how an alternating series with decreasing terms vanishing to zero converges by oscillating into its limit — and how that same oscillation bounds the truncation error. Scroll up to revisit any step.

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