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Calculus · Series-And-Sums

Sum n^(-4) for n = 1 to oo

\sum_{n=1}^{\infty} \frac{1}{n^{4}}

Step by step

  1. \sum_{n=1}^{\infty} \frac{1}{n^{4}}

    Write the sum out.

  2. 1 + \frac{1}{16} + \frac{1}{81} + \frac{1}{256} + \cdots

    The first few terms.

  3. S_{3} \approx 1.0748, S_{6} \approx 1.0811, S_{11} \approx 1.0821, S_{51} \approx 1.0823

    Partial sums approach the limit.

  4. = \frac{\pi^{4}}{90} \approx 1.0823

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

Reveal the answer
\frac{\pi^{4}}{90}