Series and sums

Partial sums, closed forms and convergence.

A sum with finitely many terms has a closed form found by pattern or induction; an infinite series converges when its partial sums approach a limit. Σ 1/n² = π²/6 is the Basel problem, Euler's first famous result.

工作范例 · sum of 1/n^2 for n = 1 to oo

Calculus · Series-And-Sums

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

\sum_{n=1}^{\infty} \frac{1}{n^{2}}

一步

\sum_{n=1}^{\infty} \frac{1}{n^{2}}
Write the sum out.
1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots
The first few terms.
S_{3} \approx 1.3611, S_{6} \approx 1.4914, S_{11} \approx 1.5580, S_{51} \approx 1.6255
Partial sums approach the limit.
= \frac{\pi^{2}}{6} \approx 1.6449
Infinite series: this converges, and the closed form is the limit of the partial sums.

发送答案

\frac{\pi^{2}}{6}

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如何: Series and sums

Write the sum out.
The first few terms.
Partial sums approach the limit.
Infinite series: this converges, and the closed form is the limit of the partial sums.

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