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
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.
Orientation and the power-series roadmap
Connect every power-series calculation to its center, radius, interval, and approximation purpose.
Power-series convergence and endpoints
Find the radius from interior behavior, then test each boundary point as a separate numerical series.
Algebra and calculus with power series
Treat algebra, differentiation, and integration as coefficient transformations with a preserved or rechecked interval.
Constructing Taylor and Maclaurin series
Match value and derivatives at one center, then separate the polynomial approximation from the infinite-series convergence claim.
Standard series and new expansions
Build new expansions from a small verified library using explicit substitutions, products, derivatives, or integrals.
Taylor bounds and approximation
Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.
Optional advanced explorations
Use coefficient recurrences and moving error envelopes to preview later analysis without weakening the core claims.
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.
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.
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.
From coefficients to a function
A power series centered at has the form
Its convergence set is an interval centered at , 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.