Chapter 21 — Further Reading

Each entry notes what to read it for and, where applicable, the exact section that parallels this chapter. The two reference texts the book is benchmarked against — Stewart and OpenStax — are mapped section-by-section first.

Mapped to This Chapter

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. - §11.2 "Series" is the direct parallel: it defines a series via partial sums, derives the geometric-series sum $a/(1-r)$, works telescoping examples, proves the harmonic series diverges, and states the divergence ($n$-th term) test — exactly the spine of our §21.2–21.6. Stewart's geometric-series application problems (bouncing ball, repeating decimals, drug accumulation) overlap our §21.10 and both case studies. - §11.3 "The Integral Test and Estimates of Sums" proves the p-series rule ($\sum 1/n^p$ converges $\iff p > 1$) that we state in §21.8 and defer the proof of to Chapter 22. - §11.4–11.5 preview the comparison, ratio, and alternating tests of our Chapter 22.

Strang, G., & Herman, E. Calculus, Volume 2. OpenStax (free, openstax.org). - §5.2 "Infinite Series" mirrors our §21.2–21.4: partial sums, geometric and telescoping series, the divergence test, and the algebraic properties of convergent series (our §21.7). - §5.3 "The Divergence and Integral Tests" contains the p-series result of our §21.8 and the integral-comparison intuition we sketch for the harmonic series in §21.5. - A genuinely free, well-illustrated alternative; its worked examples make good extra drill for the geometric and telescoping exercises.

Rigorous / Proof-Oriented Treatments

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 23 develops series with full rigor — including absolute vs. conditional convergence and the Riemann rearrangement theorem we only previewed in §21.7. Read this if you want the "Formal" rigor level behind the chapter.

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer. Chapter 2 treats series immediately after sequences, with the Cauchy criterion (our Math Major Sidebar in §21.7) proved carefully. The friendliest first real-analysis book.

Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill. Chapter 3 is the classic terse account: series, the root and ratio tests, and rearrangements. For readers who want the standard graduate-prep reference.

Apostol, T. M. (1967). Calculus, Volume I. Wiley. §§10.4–10.7 give an alternative rigorous development with a strong emphasis on the comparison principle.

On Specific Topics in the Chapter

The harmonic series and $\zeta$. du Sautoy, M. (2003). The Music of the Primes. Harper. A popular history connecting the p-series, the Riemann zeta function $\zeta(s) = \sum n^{-s}$, and the Riemann hypothesis touched on in §21.8 — no proofs, much motivation. For the technical side, Edwards, H. M. (1974). Riemann's Zeta Function (Dover) remains the standard classical reference.

Apéry's constant $\zeta(3)$. van der Poorten, A. (1979). "A proof that Euler missed… Apéry's proof of the irrationality of $\zeta(3)$." Mathematical Intelligencer, 1(4), 195–203. An unusually readable account of the result mentioned in §21.8 — that $\zeta(3)$ is irrational but has no known closed form.

Zeno's paradox. Salmon, W. C. (Ed.) (2001). Zeno's Paradoxes. Hackett. An anthology of philosophical and mathematical responses to the paradox our §21.1 and §21.10 resolve with the geometric series.

For the Case Studies

Case Study 1 (bond pricing / present value). Brealey, R. A., Myers, S. C., & Allen, F. (2019). Principles of Corporate Finance (13th ed.). McGraw-Hill. Chapters 2–3 derive the perpetuity, annuity, and Gordon-growth formulas exactly as the case study does. Bodie, Kane, & Marcus (2017). Investments (11th ed.) formalizes the price–yield relationship we computed.

Case Study 2 (drug accumulation). Rowland, M., & Tozer, T. N. Clinical Pharmacokinetics and Pharmacodynamics (4th ed.). Wolters Kluwer. The multiple-dose chapter derives the accumulation factor $1/(1-r)$, steady-state peak and trough, and loading doses in clinical units — the medical counterpart of our geometric-series computation.

A Practice Recommendation

Master two things cold. (1) The geometric-series formula $\dfrac{a}{1-r}$ for $|r| < 1$ — recognize $\sum a r^n$ instantly in finance, physics, pharmacology, and computer science, and read off the actual first term before summing. (2) The p-series dichotomy: $p > 1$ converges, $p \le 1$ diverges, with the harmonic series as the borderline failure. These two facts settle a large fraction of the series you will meet, and they anchor every test in Chapter 22.