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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.

İşlənmiş nümunə · 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}}

Addım-addım

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

    Write the sum out.

  2. 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots

    The first few terms.

  3. 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.

  4. = \frac{\pi^{2}}{6} \approx 1.6449

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

Cavabı göstər
\frac{\pi^{2}}{6}

İndi siz Problem seçin, ya da özünü yazın ya da çəkin. Hər addım, bir rəsm, cavab siz soruşana qədər gizli qalır.

Özün işlə

Pulsuz hesab hər dərsdə qeydlər əlavə edir, bitirdiyiniz işlərin qeydini saxlayır, həll etdiyiniz problemləri bir yerdə saxlayır və bu səhifə haqqında soruşa biləcəyiniz müəllimi də əlavə edir. Riyaziyyat özü daxil olub-olmadığınıza baxmayaraq hər kəsə açıqdır.

Qeyd Et Giriş

Burada istifadə edilən simvollar

Tam təsviri, şəkli və hər hərfin nə demək olduğunu görmək üçün hər hansı simvolu vurun.

Növbəti: Series and sums

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

İnsanlar soruşurlar

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

Daha çox Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests