Main content

# Series

Learn

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

Practice

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!

Learn

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)

Practice

Infinite geometric seriesGet 3 of 4 questions to level up!

Learn

nth term divergence test

Practice

nth term testGet 3 of 4 questions to level up!

Learn

Integral testWorked example: Integral test

Practice

Integral testGet 3 of 4 questions to level up!

Learn

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

Practice

p-seriesGet 3 of 4 questions to level up!

Learn

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

Practice

Direct comparison testGet 3 of 4 questions to level up!

Limit comparison testGet 3 of 4 questions to level up!

Learn

Alternating series testWorked example: alternating series

Practice

Alternating series testGet 3 of 4 questions to level up!

Learn

Ratio test

Practice

Ratio testGet 3 of 4 questions to level up!

Learn

Conditional & absolute convergence

Practice

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

Learn

Alternating series remainderWorked example: alternating series remainder

Practice

Alternating series remainderGet 3 of 4 questions to level up!

Learn

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

Practice

Taylor & Maclaurin polynomialsGet 3 of 4 questions to level up!

Learn

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

Practice

Lagrange error boundGet 3 of 4 questions to level up!

Learn

Power series introWorked example: interval of convergence

Practice

Interval of convergenceGet 3 of 4 questions to level up!

Learn

Function as a geometric seriesGeometric series as a functionPower series of arctan(2x)Power series of ln(1+x³)Geometric series interval of convergence

Practice

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

Learn

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 identity

Practice

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

Learn

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)

Practice

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!

Learn

Telescoping seriesDivergent telescoping series

Learn

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