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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}

ປັດຈຸບັນ​ທ່ານ ເລືອກ​ບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມ​ເອງ. ທຸກ​ບາດກ້າວ, ຮູບ, ຄໍາຕອບ​ຖືກ​ລົບ​ຈົນ​ກວ່າ​ເຈົ້າ​ຈະ​ຖາມ​ມັນ.

ຮັກສາ​ການ​ເຮັດວຽກ​ຂອງ​ທ່ານ​ໄວ້

ບັນຊີຟຣີ ເພີ່ມ​ຂໍ້​ມູນ​ໃສ່​ບົດຮຽນ​ທຸກ​ບົດ, ບັນທຶກ​ຂອງ​ສິ່ງທີ່​ທ່ານ​ໄດ້​ເຮັດ​ແລ້ວ, ບັນຫາ​ທີ່​ທ່ານ​ໄດ້​ແກ້​ໄຂ​ໄວ້​ໃນ​ບ່ອນ​ດຽວ ແລະ ຄູ​ສອນ​ທີ່​ທ່ານ​ສາມາດ​ຖາມ​ກ່ຽວກັບ​ໜ້າ​ນີ້. ຄະນິດສາດ​ເອງ​ແມ່ນ​ເປີດ​ໃຫ້​ທຸກ​ຄົນ, ບໍ່​ວ່າ​ຈະ​ໄດ້​ເຂົ້າ​ສູ່​ລະບົບ ຫຼື ບໍ່.

ລົງທະບຽນ ចូល

ຕົວ​ສະ​ແດງ​ທີ່​ໃຊ້​ຢູ່​ທີ່​ນີ້

ກົດ​ຕົວ​ອັກສອນ​ໃດ​ໜຶ່ງ​ເພື່ອ​ເບິ່ງ​ຄວາມ​ໝາຍ​ເຕັມ, ຮູບ ແລະ ຕົວອັກສອນ​ແຕ່ລະ​ຕົວ​ໃນ​ມັນ​ໝາຍ​ຄວາມ​ວ່າ​ແນວໃດ.

ວິທີການ: Convergence of series

  1. Does this series converge?
  2. Divergence test: if the terms do not go to 0 the series cannot converge.
  3. Ratio test: < 1 converges, > 1 diverges, = 1 says nothing.
  4. It converges, and the CAS knows the sum.

ຄໍາຖາມທີ່ຄົນຖາມ

What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

ເພີ່ມເຕີມໃນ Real Analysis

Sequences and their limitsImproper integralsTaylor approximation and errorFunctions of a complex variableContinuity and differentiability, rigorouslySequences of functions and uniform convergence