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Sequences and seriesAlgebraCollegeabout 2:23

Evaluate the sum of 1/n! from n=0 to infinity: animated lesson

A narrated, step-by-step sequences and series video lesson with captions. Final answer: The series converges (ratio test); sum ≈ 2.718281828 (matches e to 10+ digits).

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    ∑n=0∞1n!\sum_{n=0}^{\infty} \frac{1}{n!}
  2. 2. Step by step
    1. ∑n=0∞1n!\sum_{n=0}^{\infty} \frac{1}{n!}
      Write the series
    2. lim⁡n→∞1n!=0\lim_{n\to\infty} \frac{1}{n!} = 0
      nth-term test
    3. L=lim⁡n→∞1n+1=0L = \lim_{n\to\infty} \frac{1}{n+1} = 0
      Ratio test
    4. Estimate the sum numerically
  3. 3. The answer
    ∑n=0∞1n! converges≈2.718281828\sum_{n=0}^{\infty} \frac{1}{n!} \ \text{converges} \approx 2.718281828

    ratio test ⇒ converges, S ≈ 2.718281828 (matches e to 10+ digits)

  4. 4. Check
    • Partial sums (supporting evidence, not a proof): S₁₀ = 2.7182815, S₁₀₀ = 2.7182818, S₁₀₀₀ = 2.7182818, S₁₀₀₀₀ = 2.7182818

Transcript

The problem. Here is the problem: Σ 1 / n!, n equals 0 to ∞. Decide whether Σ 1 / n! (n from 0 to ∞) converges or diverges.

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

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

Check. Partial sums (supporting evidence, not a proof): S₁₀ equals 2.7182815, S₁₀₀ equals 2.7182818, S₁₀₀₀ equals 2.7182818, S₁₀₀₀₀ equals 2.7182818.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

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Evaluate the sum of 1/n! from n=0 to infinity (Lesson)