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Sequences and seriesAlgebraCollegeabout 1:59

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

    comparison with Σ 1/n ⇒ diverges

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

Transcript

The problem. Here is the problem: Σ 1 / sqrt(n), n equals 1 to ∞. Decide whether Σ 1 / sqrt(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 / sqrt(n). First terms: 1, 0.707107, 0.57735, 0.5, …. 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: 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.

The answer. So the answer is The series diverges (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 5.0209979, S₁₀₀ equals 18.589604, S₁₀₀₀ equals 61.801009, S₁₀₀₀₀ equals 198.54465.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

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