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Sequences and seriesAlgebraCollegeabout 1:59

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

A narrated, step-by-step sequences and series video lesson with captions. Final answer: The series diverges (limit comparison with Σ 1/n).

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 (0.7 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}
  2. 2. Step by step
    1. ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}
      Write the series
    2. lim⁡n→∞1n=0\lim_{n\to\infty} \frac{1}{n} = 0
      nth-term test
    3. lim⁡n→∞an1/n=1\lim_{n\to\infty} \frac{a_n}{1/n} = 1
      Limit comparison with a p-series
  3. 3. The answer
    ∑n=1∞1n diverges\sum_{n=1}^{\infty} \frac{1}{n} \ \text{diverges}

    limit comparison with Σ 1/n ⇒ diverges

  4. 4. Check
    • Partial sums (supporting evidence, not a proof): S₁₀ = 2.9289683, S₁₀₀ = 5.1873775, S₁₀₀₀ = 7.4854709, S₁₀₀₀₀ = 9.787606

Transcript

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

Step by step. Write the series. Σ a(n) for n equals 1, 2, 3, … with a(n) equals 1 / n. First terms: 1, 0.5, 0.333333, 0.25, …. nth-term test. lim a(n) equals 0, so the divergence test is inconclusive; the terms do shrink, so a finer test is needed. Limit comparison with a p-series. Positive terms: the terms shrink roughly like 1/n (estimated from a(2N)/a(N)), so compare with Σ 1/n: lim a(n) times n equals 1. The limit is a positive finite number, so the series behaves exactly like the p-series Σ 1/n, which diverges because p less than or equal to 1.

The answer. So the answer is The series diverges (limit comparison with Σ 1/n). 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 2.9289683, S₁₀₀ equals 5.1873775, S₁₀₀₀ equals 7.4854709, S₁₀₀₀₀ equals 9.787606.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

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