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
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)
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
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)
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.