Hovedindhold
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Sammenfaldende og forskellige sekvenserWorked example: sequence convergence/divergencePartial sums introPartial sums: formula for nth term from partial sumPartial sums: term value from partial sumInfinite series as limit of partial sums
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Worked example: convergent geometric seriesWorked example: divergent geometric seriesInfinite geometric series word problem: bouncing ballInfinite geometric series word problem: repeating decimalProof of infinite geometric series formulaConvergent & divergent geometric series (with manipulation)
Se hvor du er blevet bedre med disse lektioner:
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nth term divergence test
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Integral testWorked example: Integral test
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Harmonic series and 𝑝-seriesWorked example: p-seriesProof of p-series convergence criteria
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Direct comparison testWorked example: direct comparison testSammenligningstest af grænseWorked example: limit comparison testProof: harmonic series diverges
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Alternerende rækker testWorked example: alternating series
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Forholdstest
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Conditional & absolute convergence
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Alternerende rækkers restWorked example: alternating series remainder
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Taylor & Maclaurin polynomials intro (part 1)Taylor & Maclaurin polynomials intro (part 2)Worked example: Maclaurin polynomialWorked example: coefficient in Maclaurin polynomialWorked example: coefficient in Taylor polynomialVisualizing Taylor polynomial approximations
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Taylor polynomial remainder (part 1)Taylor polynomial remainder (part 2)Worked example: estimating sin(0.4) using Lagrange error boundWorked example: estimating eˣ using Lagrange error bound
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Power series introWorked example: interval of convergence
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Function as a geometric seriesGeometric series as a functionPower series of arctan(2x)Power series of ln(1+x³)Geometric series interval of convergence
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Maclaurin series of cos(x)Maclaurin series of sin(x)Maclaurin series of eˣWorked example: power series from cos(x)Worked example: cosine function from power seriesWorked example: recognizing function from Taylor seriesVisualisering af Taylor-rækketilnærmelserEuler's formula & Euler's identity
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Integrating power seriesDifferentiating power seriesFinding function from power series by integratingInterval of convergence for derivative and integralConverting explicit series terms to summation notationConverting explicit series terms to summation notation (n ≥ 2)
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Telescoping seriesDivergent telescoping series
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Formal definition for limit of a sequenceProving a sequence converges using the formal definitionFinite geometric series formulaInfinite geometric series formula intuitionProof of infinite geometric series as a limitProof of p-series convergence criteria
Test din forståelse af Rækker med disse 20 spørgsmål.

Om dette emne

Series are sums of multiple terms. Infinite series are sums of an infinite number of terms. Don't all infinite series grow to infinity? It turns out the answer is no. Some infinite series converge to a finite value. Learn how this is possible and how we can tell whether a series converges and to what value. We will also learn about Taylor and Maclaurin series, which are series that act as functions and converge to common functions like sin(x) or eˣ.