Proving ∑1/n² Converges

Real Analysis · Axiom Academy

Comparing each term to a telescoping series to bound the partial sums and force convergence. Prove that the infinite series converges. (Its exact value, , is the famous Basel problem — but here we only need to show the sum is finite, not compute it.) The partial sums climb but never reach the red bound at 2 — they settle onto π²/6. A bounded, increasing sequence must converge. Nice work — you proved converges by trapping its partial sums below a fixed number. The essential moves: Comparison strategy: when the ratio and root tests stall, bound each term by a series you can sum exactly. The key inequality: for , , because . Partial fractions: turns the bound into a telescoping series. Telescoping: , so for every N . Monotone Convergence: the partial sums are increasing and bounded above, so the series converges. The exact value: Euler later showed — the Basel problem — but bounding alone already settles convergence. This same idea generalizes: converges for every (the p -series test), one of the most-used convergence results in analysis.

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