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Here is the problem: Σ 1 / n!, n equals 0 to ∞.

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Decide whether Σ 1 / n! (n from 0 to ∞) converges or diverges.

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Write the series. Σ a(n) for n equals 0, 1, 2, … with a(n) equals 1 / n!. First terms: 1, 1, 0.5, 0.166667, ….

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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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Ratio test. Positive terms: a(n plus 1)/a(n) simplifies to 1 / (n plus 1), and its limit as n → ∞ is L equals 0. L less than 1, so the series converges (absolutely).

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Estimate the sum numerically. Partial sums S N were computed for several N and extrapolated to N → ∞ (Richardson extrapolation; alternating terms are averaged first). Two independent extrapolations agree to better than 10 to the power minus 10, giving S approximately 2.718281828. Any closed form shown is a numerical identification, not a proof.

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So the answer is The series converges (ratio test); sum approximately 2.718281828 (matches e to 10+ digits).

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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 2.7182815, S₁₀₀ equals 2.7182818, S₁₀₀₀ equals 2.7182818, S₁₀₀₀₀ equals 2.7182818.