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Sequences and seriesAlgebraCollegeabout 1:37

Evaluate the sum of 1/3^n from n=1 to infinity: animated lesson

A narrated, step-by-step sequences and series video lesson with captions. Final answer: The series converges (geometric series (|r| < 1)); sum = 1/2.

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 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    ∑n=1∞13n\sum_{n=1}^{\infty} \frac{1}{{3}^{n}}
  2. 2. Step by step
    1. ∑n=1∞13n\sum_{n=1}^{\infty} \frac{1}{{3}^{n}}
      Write the series
    2. S=a1−r=12S = \frac{a}{1-r} = \frac{1}{2}
      Recognise a geometric series
  3. 3. The answer
    ∑n=1∞13n converges=12\sum_{n=1}^{\infty} \frac{1}{{3}^{n}} \ \text{converges} = \frac{1}{2}

    geometric series (|r| < 1) ⇒ converges, S = 1/2

  4. 4. Check
    • Partial sums (supporting evidence, not a proof): S₁₀ = 0.49999153, S₁₀₀ = 0.5, S₁₀₀₀ = 0.5, S₁₀₀₀₀ = 0.5

Transcript

The problem. Here is the problem: Σ 1 over 3 ^ n, n equals 1 to ∞. Decide whether Σ 1 over 3 to the power 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 over 3 to the power n. First terms: 0.333333, 0.111111, 0.037037, 0.0123457, …. Recognise a geometric series. Each term is the previous one times r equals 1 over 3 (checked for the first 40 terms), with first term a equals 1 over 3. Since |r| less than 1 the series converges, and its sum is a/(1 minus r) equals 1 over 2 approximately 0.5.

The answer. So the answer is The series converges (geometric series (|r| less than 1)); sum equals 1 over 2. 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 0.49999153, S₁₀₀ equals 0.5, S₁₀₀₀ equals 0.5, S₁₀₀₀₀ equals 0.5.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

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Evaluate the sum of 1/3^n from n=1 to infinity (Lesson)