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Sequences and seriesAlgebraCollegeabout 1:31

Evaluate the sum of (2/3)^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 (geometric series (|r| < 1)); sum = 3.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    ∑n=0∞(23)n\sum_{n=0}^{\infty} \left({\frac{2}{3}}\right)^{n}
  2. 2. Step by step
    1. ∑n=0∞(23)n\sum_{n=0}^{\infty} \left({\frac{2}{3}}\right)^{n}
      Write the series
    2. S=a1−r=3S = \frac{a}{1-r} = 3
      Recognise a geometric series
  3. 3. The answer
    ∑n=0∞(23)n converges=3\sum_{n=0}^{\infty} \left({\frac{2}{3}}\right)^{n} \ \text{converges} = 3

    geometric series (|r| < 1) ⇒ converges, S = 3

  4. 4. Check
    • Partial sums (supporting evidence, not a proof): S₁₀ = 2.9479754, S₁₀₀ = 3, S₁₀₀₀ = 3, S₁₀₀₀₀ = 3

Transcript

The problem. Here is the problem: Σ (2 over 3) ^ n, n equals 0 to ∞. Decide whether Σ (2 over 3) to the power 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 (2 over 3) to the power n. First terms: 1, 0.666667, 0.444444, 0.296296, …. Recognise a geometric series. Each term is the previous one times r equals 2 over 3 (checked for the first 40 terms), with first term a equals 1. Since |r| less than 1 the series converges, and its sum is a/(1 minus r) equals 3.

The answer. So the answer is The series converges (geometric series (|r| less than 1)); sum equals 3. 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 2.9479754, S₁₀₀ equals 3, S₁₀₀₀ equals 3, S₁₀₀₀₀ equals 3.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

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Evaluate the sum of (2/3)^n from n=0 to infinity (Lesson)