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Sequences and seriesAlgebraCollegeabout 1:56

Evaluate the sum of (-1)^n/2^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 = 2/3.

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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. The problem
Worked example
$\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{2}^{n}}$
2. Step by step
1. $\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{2}^{n}}$
  Write the series
2. $S = \frac{a}{1-r} = \frac{2}{3}$
  Recognise a geometric series
3. The answer
$\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{2}^{n}} \ \text{converges} = \frac{2}{3}$
geometric series (|r| < 1) ⇒ converges, S = 2/3
4. Check
- Partial sums (supporting evidence, not a proof): S₁₀ = 0.66601563, S₁₀₀ = 0.66666667, S₁₀₀₀ = 0.66666667, S₁₀₀₀₀ = 0.66666667

Transcript

The problem. Here is the problem: Σ (negative 1) ^ n / 2 ^ n, n equals 0 to ∞. Decide whether Σ (negative 1) to the power n / 2 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 (negative 1) to the power n / 2 to the power n. First terms: 1, negative 0.5, 0.25, negative 0.125, …. Recognise a geometric series. Each term is the previous one times r equals negative 1 over 2 (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 2 over 3 approximately 0.6666666667.

The answer. So the answer is The series converges (geometric series (|r| less than 1)); sum equals 2 over 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 0.66601563, S₁₀₀ equals 0.66666667, S₁₀₀₀ equals 0.66666667, S₁₀₀₀₀ equals 0.66666667.

About sequences and series

Evaluate finite sums and arithmetic and geometric series.

Evaluate the sum of (-1)^n/2^n from n=0 to infinity (Lesson)