WEBVTT

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Here is the problem: Σ 1 / sqrt(n), n equals 1 to ∞.

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Decide whether Σ 1 / sqrt(n) (n from 1 to ∞) converges or diverges.

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Write the series. Σ a(n) for n equals 1, 2, 3, … with a(n) equals 1 / sqrt(n). First terms: 1, 0.707107, 0.57735, 0.5, ….

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nth-term test. lim a(n) equals 0, so the divergence test is inconclusive; the terms do shrink, so a finer test is needed.

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Limit comparison with a p-series. Positive terms: compare instead with Σ 1/n: lim a(n) times n equals ∞. The limit is ∞, so eventually the terms are larger than 1/n; that p-series diverges (p less than or equal to 1), so this one does too.

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So the answer is The series diverges (comparison with Σ 1/n).

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A series converges when its partial sums approach a finite number. The terms must tend to 0 for that to happen, but that alone is not enough (the harmonic series diverges), so a comparison, ratio, root, alternating or integral test is used to decide.

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Partial sums (supporting evidence, not a proof): S₁₀ equals 5.0209979, S₁₀₀ equals 18.589604, S₁₀₀₀ equals 61.801009, S₁₀₀₀₀ equals 198.54465.