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Calculus · Series-And-Sums

Sum (1/2)^n for n = 0 to oo

\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n}

Pass b'pass

  1. \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n}

    Write the sum out.

  2. 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots

    The first few terms.

  3. S_{3} \approx 1.7500, S_{6} \approx 1.9688, S_{11} \approx 1.9990, S_{51} \approx 2.0000

    Partial sums approach the limit.

  4. = 2

    Infinite series: this converges, and the closed form is the limit of the partial sums.

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