# Infinite sequences and series

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

## Defining convergent and divergent infinite series

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!

## Working with geometric series

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!

## The nth-term test for divergence

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!

## Integral test for convergence

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!

## Harmonic series and p-series

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!

## Comparison tests for convergence

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!

## Alternating series test for convergence

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!

## Ratio test for convergence

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!

## Determining absolute or conditional convergence

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!

## Alternating series error bound

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!

## Finding Taylor polynomial approximations of functions

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!

## Lagrange error bound

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!

## Radius and interval of convergence of power series

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!

## Finding Taylor or Maclaurin series for a function

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!

## Representing functions as power series

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!

## Optional videos

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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