Chapter 23 — Further Reading

Annotated sources for power series and Taylor series, mapped to the two reference texts this book is calibrated against (Stewart and OpenStax), then deeper analysis, computation, and the two case-study fields.

Primary textbook mapping

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. The chapter's closest companion. Map our sections to Stewart's Chapter 11 as follows:

  • §11.8 Power Series ↔ our Section 23.2. Radius and interval of convergence via the ratio test, with the same endpoint-testing discipline.
  • §11.9 Representations of Functions as Power Series ↔ our Sections 23.3 and 23.6. Term-by-term differentiation and integration; building $\ln(1+x)$ and $\arctan x$ from the geometric series.
  • §11.10 Taylor and Maclaurin Series ↔ our Sections 23.4, 23.5, and 23.10. The coefficient formula, the seven standard series, Taylor's remainder, and the question of when a series equals its function. Stewart's binomial-series treatment matches our Section 23.4.
  • §11.11 Applications of Taylor Polynomials ↔ our Sections 23.5 and 23.8. Error estimation and the relativistic kinetic-energy example (our Case Study 1).

Strang, G., & Herman, E. Calculus, Volume 2. OpenStax (free, openly licensed). Exceeds many texts on application breadth.

  • §6.1 Power Series and Functions ↔ our Section 23.3.
  • §6.2 Properties of Power Series ↔ our Sections 23.2 and 23.6 (radius, term-by-term operations, the Cauchy product).
  • §6.3 Taylor and Maclaurin Series ↔ our Sections 23.4–23.5.
  • §6.4 Working with Taylor Series ↔ our Sections 23.6–23.8. Contains the binomial series, the $e^{-x^2}$ integration (our Section 23.7 anchor and Case Study 2), and the relativity application — the single best free parallel to this chapter. Read it after the index.

Our internal Stewart and OpenStax cross-reference appendices (appendix-h-stewart-chapter-mapping.md and appendix-i-openstax-chapter-mapping.md) give the full book-wide table.

Rigorous analysis (for the Math Major Sidebars)

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 24 ("Approximation by Polynomial Functions") and Chapter 27 ("Power Series") prove Taylor's theorem and the radius-of-convergence trichotomy in full, including the $e^{-1/x^2}$ counterexample of our Section 23.10. The gold standard for the why.

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. Chapter 8 develops power series with uniform convergence, justifying the term-by-term calculus we use freely in Section 23.3. Terse but definitive.

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer. Chapter 6 is the gentlest rigorous route to the same material — uniform convergence, the Weierstrass M-test, and why differentiating a series term-by-term is legal. Ideal bridge from this chapter to a first analysis course.

The complex-plane explanation of the radius (Section 23.9)

Needham, T. (1997). Visual Complex Analysis. Oxford University Press. Beautifully illustrates why a real Taylor series' radius is set by complex singularities — the "invisible poles at $\pm i$" behind $1/(1+x^2)$. The most intuitive treatment of Section 23.9's central fact.

Computation of elementary and special functions

Muller, J.-M. (2016). Elementary Functions: Algorithms and Implementation (3rd ed.). Birkhäuser. The definitive account of how libm actually computes $\sin$, $\exp$, $\ln$ — range reduction, polynomial (Taylor/Chebyshev) cores, and error analysis. The engineering reality behind Section 23.5's "this is how calculators work."

Press, W. H., et al. (2007). Numerical Recipes (3rd ed.). Cambridge University Press. Practical chapters on series evaluation, the error function, and when to abandon a convergent series for an asymptotic one (our Section 23.7 Computational Note).

Case Study 1 — special relativity and GPS

Taylor, E. F., & Wheeler, J. A. (1992). Spacetime Physics (2nd ed.). W. H. Freeman. Derives $E_k = (\gamma-1)mc^2$ and expands it to recover $\tfrac12 mv^2$, exactly the binomial-series argument of our Section 23.8.

Ashby, N. (2003). "Relativity in the Global Positioning System." Living Reviews in Relativity, 6, 1. Openly available; the authoritative source for why the Lorentz-factor correction (the second Taylor term) is mandatory in GPS, and the origin of the microseconds-per-day figure in Case Study 1.

Case Study 2 — the normal distribution and special functions

Abramowitz, M., & Stegun, I. (Eds.) (1965). Handbook of Mathematical Functions. Dover — and its modern successor, the NIST Digital Library of Mathematical Functions (DLMF), https://dlmf.nist.gov/. The reference for the error-function series (our Section 23.7) and its asymptotic tail expansion.

Cody, W. J. (1969). "Rational Chebyshev approximations for the error function." Mathematics of Computation, 23, 631–637. What real libraries (scipy.special.erf) use once the naive Taylor series loses accuracy — the next step beyond Case Study 2.

Padé approximants (a sequel to Taylor polynomials)

Baker, G. A., & Graves-Morris, P. (1996). Padé Approximants (2nd ed.). Cambridge University Press. Rational-function approximations that often beat a Taylor polynomial of the same total degree — the natural follow-on once you have mastered truncation and error bounds.

Computer algebra (verify your hand work)

Visualization

3Blue1Brown, "Taylor series" (YouTube). The clearest animated intuition for "every smooth function is locally a polynomial," complementing Figure 23.1.

A practice recommendation

Memorize the seven standard series of Section 23.4 cold — they appear in every application of the method. Then re-derive the dependent ones to lock in the manipulations: get $\cos x$ from $\sin x$ by differentiation, $\ln(1+x)$ from $1/(1+x)$ by integration, and $\arctan x$ from $1/(1+x^2)$ by integration. Finally, integrate $e^{-x^2}$ term-by-term once by hand (Section 23.7); that single exercise contains the whole chapter in miniature.