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Sequences & Series Articles: Convergence Tests, Taylor and Laurent Series

Short articles that answer one question about sequences and series at a time, from the geometric series formula and a flowchart for choosing a convergence test to the Taylor and Laurent series used in engineering. Each has worked numbers and links to the full lessons.

Sequences & series How to Choose a Convergence Test: an 8-Step Flowchart with an Example at Every Step Pick the right convergence test the first time: an 8-step flowchart (n-th term, geometric, p-series, telescoping, limit comparison, ratio, root,… Read the article → Sequences & series What Is a Laurent Series and How Is It Different from a Taylor Series? (Explained with Pictures) Laurent series vs Taylor series: negative powers, the principal part, the annulus of convergence, types of singularity, residues and the link to… Read the article → Sequences & series How to Find the Radius of Convergence: 3 Methods, Endpoint Checks and the Complex-Plane Reason Three ways to find the radius of convergence of a power series (ratio test, root test, nearest complex singularity), plus the endpoint checks for… Read the article → Sequences & series What Do You Learn in Sequences and Series? A Roadmap from Sequences to Laurent Series The whole sequences and series chapter on one page: what each topic answers, its key formula and typical exam questions, from sequences and… Read the article → Sequences & series Geometric Series: Formula, Examples and How to Find the Sum Geometric series formula for the sum of n terms and the infinite sum a₁/(1 − r) when |r| < 1, with the derivation, step-by-step examples, common… Read the article → Sequences & series Mathematical Induction Examples with Step-by-Step Solutions, Including Strong Induction 14 worked proof-by-induction examples: writing the induction statement P(n), sums, divisibility, inequalities, recurrences and strong induction,… Read the article → Sequences & series What Is a Taylor Series, What Is It Used For, with Examples What a Taylor series is, the Maclaurin series table to memorise, and 8 worked examples: approximation, expanding about other points, how many… Read the article →

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Go deeper in the lessons

Sequences & series, 12 pages (Thai) → Sequence & series calculators with steps (Thai) Strong induction, 17 examples (Thai) All articles