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# Infinite sequences and series

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑8 (EU)

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.1 (EK), LIM‑7.A.2 (EK)

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Convergent and divergent sequencesWorked 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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Sequence convergence/divergenceGet 3 of 4 questions to level up!

Partial sums introGet 3 of 4 questions to level up!

Partial sums & seriesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.3 (EK), LIM‑7.A.4 (EK)

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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 formula

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Infinite geometric seriesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.5 (EK)

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nth term divergence test

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nth term testGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.6 (EK)

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Integral testWorked example: Integral test

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Integral testGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.7 (EK)

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Harmonic series and 𝑝-seriesWorked example: p-seriesProof of p-series convergence criteria

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p-seriesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.8 (EK), LIM‑7.A.9 (EK)

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Direct comparison testWorked example: direct comparison testLimit comparison testWorked example: limit comparison testProof: harmonic series diverges

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Direct comparison testGet 3 of 4 questions to level up!

Limit comparison testGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.10 (EK)

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Alternating series testWorked example: alternating series

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Alternating series testGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.11 (EK)

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Ratio test

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Ratio testGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.A (LO), LIM‑7.A.12 (EK), LIM‑7.A.13 (EK)

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Conditional & absolute convergence

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Determine absolute or conditional convergenceGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑7 (EU), LIM‑7.B (LO), LIM‑7.B.1 (EK)

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Alternating series remainderWorked example: alternating series remainder

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Alternating series remainderGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑8 (EU), LIM‑8.A (LO), LIM‑8.A.1 (EK), LIM‑8.A.2 (EK), LIM‑8.B (LO), LIM‑8.B.1 (EK)

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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 & Maclaurin polynomialsGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑8 (EU), LIM‑8.C (LO), LIM‑8.C.1 (EK)

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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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Lagrange error boundGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑8 (EU), LIM‑8.D (LO), LIM‑8.D.1 (EK), LIM‑8.D.2 (EK), LIM‑8.D.3 (EK), LIM‑8.D.4 (EK)

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Power series introWorked example: interval of convergence

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Interval of convergenceGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑8 (EU), LIM‑8.E (LO), LIM‑8.E.1 (EK), LIM‑8.F (LO), LIM‑8.F.1 (EK), LIM‑8.F.2 (EK)

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Function as a geometric seriesGeometric series as a functionPower series of arctan(2x)Power series of ln(1+x³)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 seriesVisualizing Taylor series approximationsEuler's formula & Euler's identityGeometric series interval of convergence

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Function as a geometric seriesGet 3 of 4 questions to level up!

Maclaurin series of sin(x), cos(x), and eˣGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑8 (EU), LIM‑8.D (LO), LIM‑8.D.6 (EK), LIM‑8.G (LO), LIM‑8.G.1 (EK)

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Integrating power seriesDifferentiating power seriesFinding function from power series by integratingInterval of convergence for derivative and integral

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Integrate & differentiate power seriesGet 3 of 4 questions to level up!

Integrals & derivatives of functions with known power seriesGet 3 of 4 questions to level up!

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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

### About this unit

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.AP® is a registered trademark of the College Board, which has not reviewed this resource.