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Infinite series, power series and Taylor series

Calculus II: integration and infinite series · Lesson 4 of 4

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Lesson 4 of 4 · Calculus II: integration and infinite series

Infinite series, power series and Taylor series

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By the end you’ll be able to

Determine convergence with the divergence, comparison, limit comparison, ratio, root, integral and alternating series tests, and distinguish absolute from conditional convergence
Find the radius and interval of convergence of a power series, including behaviour at the endpoints

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Derive Taylor and Maclaurin series from the standard series for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $\arctan x$, $1/(1-x)$ and $(1+x)^\alpha$ by substitution and term-by-term differentiation or integration

Sum series in closed form: geometric, telescoping, and known power series evaluated at a point (e.g. $\sum n x^n$, $\sum x^n/n!$)

Use Taylor polynomials to approximate values, find limits and specific derivatives at a point, and bound the error with the Lagrange remainder or the alternating series estimate

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