Chapter 22 — Further Reading

Annotated, with explicit section mapping to the two reference texts this book is measured against (continuity §8). Read the strategy sections last — they are where the chapter's real skill lives.

Primary Textbook Mapping

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage. The closest match to this chapter's scope and exercise depth. Map our sections onto Stewart's Chapter 11: - §11.3 — the Integral Test and $p$-series → our §22.2. Includes Stewart's remainder estimate for the integral test, a nice complement to our alternating bound. - §11.4 — the Comparison Tests (direct and limit) → our §22.3–22.4. - §11.5 — Alternating Series and the Alternating Series Estimation Theorem → our §22.7 (same error bound $|L-S_N|\le b_{N+1}$). - §11.6 — the Ratio and Root Tests, plus absolute vs. conditional convergence and rearrangements → our §22.5, §22.6, §22.8. - §11.7 — "Strategy for Testing Series" → our decision framework, §22.10. Work Stewart's mixed exercises here; they are the best available drill for test selection.

Strang, G., & Herman, E. Calculus, Volume 2. OpenStax (free online). Map onto OpenStax Chapter 5: - §5.3 — Divergence and Integral Tests → our §22.1 (divergence), §22.2 (integral). - §5.4 — Comparison Tests → our §22.3–22.4. - §5.5 — Alternating Series (with the remainder estimate and absolute/conditional convergence) → our §22.7–22.8. - §5.6 — Ratio and Root Tests, and the "choosing a test" student project → our §22.5–22.6, §22.10. OpenStax's worked examples and free interactive checkpoints are an excellent first pass before tackling Stewart's harder exercises.

Rigorous Analysis (for the Math Major Sidebars)

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 23 ("Infinite Series") proves every test in this chapter from first principles, including the absolute-convergence theorem and rearrangements. The cleanest route to the why behind §22.5–22.8.

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer. §2.7 develops the series tests with unusually readable proofs, and §2.1 motivates the whole subject with the Riemann rearrangement theorem — directly extends our §22.8 Math Major Sidebar.

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. Chapter 3 is the terse, authoritative reference; its treatment of the root test via $\limsup$ justifies the "root test is strictly stronger" claim in our §22.6 sidebar (the Cauchy–Hadamard formula).

Bartle, R. G., & Sherbert, D. R. (2011). Introduction to Real Analysis (4th ed.). Wiley. §3.7 and §9.1–9.2; a gentler companion to Rudin with many solved examples.

Certified Computation and Convergence Rate (Case Study 1)

Muller, J.-M. (2016). Elementary Functions: Algorithms and Implementation (3rd ed.). Birkhäuser. Exactly the engineering of Case Study 1: how real math libraries evaluate log/exp with proven accuracy, including argument reduction (to keep the ratio $\rho$ small) and series selection. Read after §22.5 and §22.7.

Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. The definitive reference on why convergence rate and summation order govern real accuracy — the practical face of absolute vs. conditional convergence (§22.8).

Borwein, J., & Borwein, P. (1987). Pi and the AGM. Wiley. A tour of fast and slow series for classical constants ($\pi$, $\ln 2$), with rate analysis front and center — the perfect sequel to "why the Leibniz series is a terrible way to compute $\pi$" (Exercise 22.25).

Expected Value as a Series (Case Study 2)

Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 (3rd ed.). Wiley. Treats the St. Petersburg paradox and the existence of expectations explicitly as convergence questions — the rigorous backdrop to Case Study 2.

Bernoulli, D. (1738; trans. Sommer, 1954). "Exposition of a New Theory on the Measurement of Risk." Econometrica, 22, 23–36. The original expected-utility resolution of St. Petersburg: replacing the payoff with its logarithm turns a divergent series into a convergent one.

Taleb, N. N. (2020). Statistical Consequences of Fat Tails. STEM Academic Press (open access). A modern applied account of infinite-variance distributions — the $p=1$ boundary of §22.2.1 made vivid, and why it breaks standard statistics.

The Riemann Rearrangement Theorem (§22.8)

Galanor, S. (1987). "Riemann's Rearrangement Theorem." Mathematics Teacher, 80(9), 675–681. An accessible, picture-driven exposition suitable right after finishing §22.8.

Apostol, T. M. (1967). Calculus, Volume I (2nd ed.). Wiley. §10.20–10.21 give a careful, classical proof of the rearrangement theorem alongside the full battery of tests.

A Practice Recommendation

The single most useful exercise is to internalize the decision framework (§22.10): given any series, you should name the deciding test within seconds. Drill on Stewart §11.7 and OpenStax §5.6's mixed problems — roughly 40–50 varied series — until pattern recognition is automatic. That fluency is exactly what Chapter 23's power-series radius-of-convergence questions will demand.