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Calculus · Series-And-Sums

Sum 2^(-k) for k = 1 to oo

\sum_{k=1}^{\infty} 2^{- k}

Step by step

  1. \sum_{k=1}^{\infty} 2^{- k}

    Write the sum out.

  2. \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots

    The first few terms.

  3. S_{3} \approx 0.87500, S_{6} \approx 0.98438, S_{11} \approx 0.99951, S_{51} \approx 1.0000

    Partial sums approach the limit.

  4. = 1

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

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