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Convergence of series

Divergence, ratio, comparison and integral tests — and the harmonic series.

An infinite series converges when its partial sums approach a limit. The terms going to zero is necessary but not enough — the harmonic series 1 + ½ + ⅓ + … diverges even though its terms vanish. The ratio test settles most series with factorials or powers; comparison and the integral test handle the p-series family.

ຕົວຢ່າງ​ທີ່​ໄດ້​ເຮັດ: does 1/n^2 for n = 1 to oo converge

Calculus · Series-And-Sums

Convergence of n^(-2)

\frac{1}{n^{2}}

ຂັ້ນ​ຕອນ​ຕໍ່​ຂັ້ນ​ຕອນ

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

    Does this series converge?

  2. \lim_{n\to\infty} \frac{1}{n^{2}} = 0

    Divergence test: if the terms do not go to 0 the series cannot converge.

  3. \lim \left|\frac{a_{k+1}}{a_k}\right| = \lim \left|{\frac{n^{2}}{\left(n + 1\right)^{2}}}\right| = 1

    Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.

  4. = \frac{\pi^{2}}{6}

    It converges, and the CAS knows the sum.

ເປີດ​ເຜີຍ​ຄຳ​ຕອບ
\text{converges to } \frac{\pi^{2}}{6}

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ເພີ່ມເຕີມໃນ Analysis

Sequences and their limitsImproper integralsTaylor approximation and errorFunctions of a complex variable