The Harmonic Series Diverges

Real Analysis · Axiom Academy

EXAMPLE The Harmonic Series Diverges A classic proof using strategic grouping to show the partial sums are unbounded. Show that the harmonic series diverges — even though its terms . We will group the terms so that each successive block sums to more than , forcing the partial sums to grow without bound. Each block holds twice as many terms as the one before, yet every block still totals more than 1/2 — so the running total climbs past 1/2, 1, 3/2, 2, … without limit. Nice work — you've reconstructed Oresme's classic proof that the harmonic series diverges. The moves to remember: Strategic grouping is the whole trick: group the terms so each block runs from to . The k -th block has 2^k terms: each successive block holds twice as many terms as the last. Every block beats : replacing each term by the block's smallest gives , so the true sum is even larger. That forces the partial sums up: after k blocks, , which grows without bound. Terms shrinking to zero is not enough: , yet the series still diverges — a foundational warning in analysis. Nicole Oresme found this argument in the 14th century. It shows that convergence demands more than terms heading to zero — it's the rate that decides.

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