Absolute vs Conditional Convergence

Real Analysis · Axiom Academy

LESSON Absolute vs Conditional Convergence Some series survive taking absolute values; others converge only by a delicate cancellation. Watch the difference. 1. Absolute Convergence — Both Totals Settle A series is absolutely convergent when the series of absolute values converges. Take the alternating geometric series below. The signed total steps up and down, while the absolute total only ever climbs. Watch where each one ends up. Take absolute values — it's just a geometric series The signed total settles at , and the absolute total settles at . Because is finite, the series is absolutely convergent . 2. Conditional Convergence — The Split Now the alternating harmonic series. The signed total homes in on , exactly as in Step 1. But strip the signs and the absolute total becomes the harmonic series — and it climbs off the top of the frame forever. The two totals split apart : one converges, the other diverges. Take absolute values — now it's the harmonic series The signed total reaches by the Alternating Series Test, yet . A series that converges while diverges is conditionally convergent . 3. Why Absolute Convergence Is Stronger The two ideas are tied together by one theorem. Every term is trapped between -|a_n| and +|a_n| . If the outer band converges, that band squeezes the partial sums of into convergence too — there is no room left to wander. Watch the band close in.

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