maths.free › Calculus › Series and sums

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.

ምሳሌ · 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}}

እርምጃ በደረጃ

  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.

መልሱን አሳይ
\frac{\pi^{2}}{6}

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: 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.

ጥያቄዎች

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.

በ 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