Riemann Rearrangement Theorem

Real Analysis · Axiom Academy

LESSON Riemann Rearrangement Theorem For a conditionally convergent series, the order of the terms isn't a detail — it decides the sum, and you can steer it to any value you like. A series is absolutely convergent if also converges, and conditionally convergent if it converges but diverges. The animation runs the partial sums of both at once: the geometric series settles to a fixed height no matter how you shuffle it, while the alternating harmonic series only just hangs together — and that fragility is exactly what Riemann exploits. Rearrangement preserves the sum. Rearrangement can change the sum. 2. The Alternating Harmonic Series The standard example of conditional convergence is the alternating harmonic series. Left in its natural order it converges to , but the series of absolute values is the harmonic series, which diverges. The animation plots its partial sums: they oscillate above and below the dashed line and squeeze in on . By the alternating series test, an alternating sum whose terms shrink monotonically to 0 converges. The terms , so the partial sums settle — but only conditionally, because dropping the signs gives the divergent . 3. The Engine: Both Halves Diverge

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