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Calculus II · Unit 4B · hub

Unit 4B: Power Series and Taylor Series

Learn power series, intervals of convergence, Taylor polynomials, Taylor series, remainder bounds, and approximation through rigorous visual lessons.

Core textbook pathChoose the next lesson from the ordered unit map.

Core textbook

The complete Unit 4B path

Begin by treating power series as functions, determine exact convergence intervals, build Taylor expansions from derivative data, and finish with certified approximation bounds.

What this unit teaches

Turn infinite series into controlled function models.

Find power-series radii and endpoints; differentiate, integrate, and combine series; construct Taylor and Maclaurin expansions; transform standard series; and certify approximation error with remainder bounds.

Prerequisites

Unit 4A convergence tests plus derivative and integral fluency.

You should classify numerical series, apply ratio and alternating-series tests, manipulate factorials and powers, compute repeated derivatives, and use basic definite integrals. Return to Unit 4A whenever endpoint testing exposes a convergence gap.

Section

Orientation and the power-series roadmap

Connect every power-series calculation to its center, radius, interval, and approximation purpose.

  1. 01Unit 4B: Power Series and Taylor Serieshub
Section

Power-series convergence and endpoints

Find the radius from interior behavior, then test each boundary point as a separate numerical series.

  1. 02Power Series as Functionslesson
  2. 03Radius and Interval of Convergencelesson
  3. 04Using the Ratio Test on Power Serieslesson
  4. 05Endpoint Testing for Power Serieslesson
Section

Algebra and calculus with power series

Treat algebra, differentiation, and integration as coefficient transformations with a preserved or rechecked interval.

  1. 06Algebra with Power Serieslesson
  2. 07Differentiating and Integrating Power Serieslesson
  3. 08The Geometric Series as a Function Librarylesson
Section

Constructing Taylor and Maclaurin series

Match value and derivatives at one center, then separate the polynomial approximation from the infinite-series convergence claim.

  1. 09Taylor Polynomialslesson
  2. 10Maclaurin Serieslesson
  3. 11Taylor Series Centered at alesson
  4. 12Standard Maclaurin Serieslesson
  5. 13Logarithm and Arctangent Serieslesson
Section

Standard series and new expansions

Build new expansions from a small verified library using explicit substitutions, products, derivatives, or integrals.

  1. 14Building New Series from Known Oneslesson
  2. 15The Binomial Serieslesson
  3. 16Taylor's Theorem and the Remainderlesson
Section

Taylor bounds and approximation

Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.

  1. 17Alternating-Series Approximation for Taylor Serieslesson
  2. 18Using Series to Approximate Definite Integralslesson
  3. 19Small-Angle Approximations in Physicslesson
  4. 20A Preview of Series Solutions to Differential Equationslesson
Section

Optional advanced explorations

Use coefficient recurrences and moving error envelopes to preview later analysis without weakening the core claims.

  1. 21Why Termwise Operations Need Uniform Controllesson

Practice around the path

Reviews, quizzes, diagnostics, and exams

Use these after a section or whenever a worked example reveals a specific gap. The answer keys are separated so an honest first attempt stays easy.

reviewPower Series ReviewReview power series review with connected concepts, test-selection guidance, worked reasoning, and exam preparation.reviewTaylor Series ReviewReview taylor series review with connected concepts, test-selection guidance, worked reasoning, and exam preparation.quizUnit 4B Concept QuizCheck your understanding with the Unit 4B Concept Quiz, covering definitions, theorem hypotheses, strategy, and common misconceptions.practiceUnit 4B Mixed PracticePractice unit 4b mixed practice with structured problems that require method selection, justification, and error control.examUnit 4B Practice Exam AAttempt Unit 4B Practice Exam A under exam conditions, then use the separate worked answer key to diagnose gaps.examUnit 4B Practice Exam BAttempt Unit 4B Practice Exam B under exam conditions, then use the separate worked answer key to diagnose gaps.referenceUnit 4B Reference SheetUse Unit 4B Reference Sheet as a concise reference for formulas, theorem hypotheses, decisions, and common errors.referenceCommon Errors in Power and Taylor SeriesUse Common Errors in Power and Taylor Series as a concise reference for formulas, theorem hypotheses, decisions, and common errors.

Check your work

Published exam answer keys

Every exam has a separately routed, numbered key. Finish the exam first, then compare one item at a time.

Go deeper

Focused series explorations

These articles zoom in on one interval decision, Taylor construction, or certified approximation. They are enrichment around the textbook path, not a replacement for it.

Repair prerequisite gaps

Return easily to Unit 4A convergence foundations

When endpoint testing or a numerical-series argument is the obstacle, review the matching Unit 4A lesson, then return here to finish the power-series claim.

Review Unit 4A convergence →

Unit 4B: Power Series and Taylor Series

A numerical series produces one number. A power series contains a variable and therefore produces a function wherever the series converges. This change turns convergence testing into a form of function construction: a single infinite expression may represent a familiar function on one interval and diverge outside it.

Taylor series take the idea further. Derivatives at one center encode local behavior, and a polynomial built from that derivative data can approximate the function. Under suitable control of the remainder, increasing-degree polynomials converge to the function itself. The unit develops the standard series library, interval and radius calculations, termwise calculus, Taylor error bounds, and practical approximation examples.

Concept

From coefficients to a function

A power series centered at aa has the form

∑n=0∞cn(x−a)n.\sum_{n=0}^{\infty}c_n(x-a)^n.

Its convergence set is an interval centered at aa, possibly including neither, one, or both endpoints.

Unit map

• Power series, radius, interval, and endpoint testing • Algebra and calculus with power series • Geometric-series transformations • Taylor and Maclaurin polynomials • Standard series and construction techniques • Binomial series • Remainders, error bounds, and numerical applications • A preview of series in physics and differential equations

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.