Analyzing ∑(-1)ⁿ/n

Real Analysis · Axiom Academy

Showing the alternating harmonic series converges conditionally, but not absolutely. Classify the convergence of the alternating harmonic series . Does it converge? If so, is the convergence absolute or conditional ? Partial sums settling on the limit The partial sums zig-zag above and below the limit, and the swings shrink — that shrinking-overshoot pattern is exactly what the Alternating Series Test guarantees. The series settles on . Nice work. You classified the alternating harmonic series: it converges by the Alternating Series Test, yet its absolute-value series diverges — the textbook case of conditional convergence. Alternating Series Test: if the magnitudes are positive, decreasing, and , then converges. Absolute vs. conditional: a series is absolutely convergent when converges; if converges but diverges, the convergence is conditional . The harmonic series diverges: , so cannot converge. Result: is conditionally convergent, with value . The sign cancellation from the alternating terms is what rescues convergence — drop the signs and the same series blows up. That gap between and is the whole point of conditional convergence.

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