Chapter 21 — Exercises

38 problems on partial sums, geometric series, telescoping series, the harmonic and p-series, the divergence test, repeating decimals, and applications. Difficulty is marked ⭐ (warm-up) to ⭐⭐⭐⭐ (challenge). Work the lower tiers until they are automatic before reaching for the harder ones — the foundations here are used in every one of Chapters 22, 23, and 24.

Selected answers appear in appendices/answers-to-selected.md.

Part A — Partial Sums and the Definition (§21.2)

21.1 (⭐) The partial sums of a series are $S_N = \dfrac{2N}{N+3}$. Does the series converge, and if so, to what value?

21.2 (⭐) For the series in 21.1, recover the term $a_4 = S_4 - S_3$.

21.3 (⭐⭐) A series has partial sums $S_N = \dfrac{N^2}{N+1}$. Use the divergence-of-partial-sums criterion to decide whether it converges, and find a general formula for $a_n = S_n - S_{n-1}$ (for $n \ge 2$).

21.4 (⭐⭐) Write out the first four partial sums $S_1, S_2, S_3, S_4$ of $\displaystyle\sum_{n=1}^\infty (-1)^{n+1}$ and explain, in one sentence, why the series diverges even though the terms do not run off to infinity.

21.5 (⭐⭐) For $a_n = \left(\tfrac{1}{2}\right)^n$, verify by writing out $S_1, S_2, S_3$ that $S_N = 1 - 2^{-N}$, then state the limit. Which type of divergence/convergence behavior does this illustrate?

Part B — Geometric Series (§21.3)

21.6 (⭐) $\displaystyle\sum_{n=0}^\infty \left(\tfrac{2}{3}\right)^n$.

21.7 (⭐) $\displaystyle\sum_{n=1}^\infty \left(-\tfrac{1}{4}\right)^n$.

21.8 (⭐) $\displaystyle\sum_{n=0}^\infty 2^n$ — converge or diverge? State the value of $r$ and the reason.

21.9 (⭐⭐) $\displaystyle\sum_{n=0}^\infty 5\,(0.99)^n$. (Slowly converging — give the exact sum.)

21.10 (⭐⭐) $\displaystyle\sum_{n=2}^\infty 3\left(\tfrac{1}{2}\right)^n$ — geometric starting at $n = 2$. Identify the actual first term before summing.

21.11 (⭐⭐) $\displaystyle\sum_{n=1}^\infty \frac{2^n + 3^n}{6^n}$. (Split into two geometric series via §21.7 linearity.)

21.12 (⭐⭐) Sum the series $1 + 0.5 + 0.25 + 0.125 + \cdots$ to infinity.

21.13 (⭐⭐⭐) For which values of $x$ does $\displaystyle\sum_{n=0}^\infty (x-2)^n$ converge, and what is the sum as a function of $x$ on that interval? (A first glimpse of the power series of Chapter 23.)

Part C — Repeating Decimals (§21.3)

21.14 (⭐) Express $0.\overline{6}$ as a fraction in lowest terms.

21.15 (⭐⭐) Express $0.\overline{142857}$ as a fraction and confirm it equals $\tfrac{1}{7}$.

21.16 (⭐⭐) Show that $0.\overline{9} = 1$ exactly, using the geometric-series sum (not the "$10x$" algebra trick).

21.17 (⭐⭐⭐) Express the eventually-repeating decimal $0.4\overline{12}$ as a fraction. (Split into a non-repeating part plus a shifted geometric tail.)

Part D — Telescoping Series (§21.4)

21.18 (⭐⭐) $\displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)}$. (Partial fractions, then telescope.)

21.19 (⭐⭐) $\displaystyle\sum_{n=1}^\infty \frac{2}{(n+1)(n+3)}$. (Factor, partial-fraction; note the offset.)

21.20 (⭐⭐⭐) $\displaystyle\sum_{n=2}^\infty \frac{1}{n^2 - 1}$. (Factor $n^2 - 1 = (n-1)(n+1)$; watch for two surviving leading terms.)

21.21 (⭐⭐⭐) $\displaystyle\sum_{n=1}^\infty \left(\sqrt{n+1} - \sqrt{n}\right)$. Show the partial sum is $\sqrt{N+1} - 1$ and conclude that the series diverges, despite its terms shrinking to zero — a telescoping echo of the harmonic warning.

21.22 (⭐⭐⭐) $\displaystyle\sum_{n=1}^\infty \ln\!\left(\frac{n}{n+1}\right)$. Write the term as $\ln n - \ln(n+1)$, telescope the partial sum, and decide convergence.

Part E — Harmonic Series and p-Series (§21.5, §21.8)

21.23 (⭐⭐⭐⭐) Reproduce the grouping (Oresme) proof from §21.5 that $\displaystyle\sum_{n=1}^\infty \tfrac1n$ diverges. State clearly why each doubling block exceeds $\tfrac12$.

21.24 (⭐⭐⭐) Compute the harmonic partial sums $S_{10}$, $S_{100}$, $S_{1000}$ numerically and compare each to $\ln N + \gamma$ with $\gamma \approx 0.5772$.

21.25 (⭐) Does $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$ converge or diverge? State the $p$-value and the rule from §21.8. (Its value is $\pi^2/6 \approx 1.6449$.)

21.26 (⭐⭐) Decide convergence for each, citing the $p$-series rule: (a) $\sum \tfrac{1}{n^{3/2}}$, (b) $\sum \tfrac{1}{\sqrt n}$, (c) $\sum \tfrac{5}{n^2}$.

21.27 (⭐⭐⭐) Explain why $\displaystyle\sum_{n=1}^\infty \frac{1}{2n}$ diverges, using the constant-multiple property of §21.7 and the harmonic series — without any test from Chapter 22.

21.28 (⭐⭐⭐) Does $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{1.001}}$ converge? State the $p$-value and the rule; comment on how its slow convergence would mislead a purely numerical experiment of $10^6$ terms.

Part F — The Divergence Test and Its Caveat (§21.6)

21.29 (⭐) Apply the divergence ($n$-th term) test to each, stating "diverges" or "test inconclusive": (a) $\sum \dfrac{n}{n+1}$, (b) $\sum (-1)^n$, (c) $\sum \dfrac{1}{n^2}$, (d) $\sum \cos n$.

21.30 (⭐⭐) For $\displaystyle\sum_{n=1}^\infty \frac{3n^2 + 1}{5n^2 + n}$, evaluate $\lim_{n\to\infty} a_n$ and decide whether the divergence test applies.

21.31 (⭐⭐⭐⭐) The series $\displaystyle\sum_{n=1}^\infty \frac{n^2}{n^3 + 1}$ has terms $a_n \to 0$, yet the divergence test is silent. Explain precisely why the test cannot settle this case, and state — citing §21.6 — the single sentence you must never write.

21.32 (⭐⭐) Give one series for which the divergence test is useful (settles the question) and one for which it is useless (cannot settle it). Justify each in a sentence.

Part G — Applications (§21.10) — at least two fields

21.33 (⭐⭐⭐, economics) Present value of a perpetuity. A perpetuity pays \$500 per year forever at a 4% annual discount rate. Find its present value using $PV = C/r$, and identify the common ratio of the underlying geometric series.

21.34 (⭐⭐⭐, physics) Bouncing ball. A ball is dropped from $h = 10$ m and rebounds to a fraction $r = 0.6$ of its previous height on each bounce. Find the total vertical distance it travels using $d = h\,\dfrac{1+r}{1-r}$.

21.35 (⭐⭐⭐, medicine) Drug accumulation. A patient takes a fixed dose $D = 200$ mg on a fixed schedule, and a fraction $r = 0.25$ of each dose remains in the body when the next is taken. Find the long-run steady-state body load using $D/(1-r)$.

21.36 (⭐⭐⭐, economics/macro) The multiplier effect. A government injects \$1 billion into an economy. Each recipient spends a fraction $c = 0.8$ (the marginal propensity to consume) of what they receive, and so on, forever. Find the total increase in spending as a geometric series, and compare it to the multiplier $1/(1-c)$.

21.37 (⭐⭐⭐⭐, geometry/physics) Cantor set. Starting from $[0,1]$, remove the open middle third; from each remaining interval remove its middle third; repeat forever. Express the total length removed as a geometric series and evaluate it. What does the result say about the length of the leftover Cantor set?

Part H — Synthesis and Reflection

21.38 (⭐⭐⭐⭐) Why does $\sum 1/n$ diverge while $\sum 1/n^2$ converges, even though both have terms tending to zero? Give the geometric/integral-comparison intuition from §21.5 and §21.8, and connect it to the boundary case $p = 1$.

Total: 38 exercises. Tier counts: ⭐ ×8, ⭐⭐ ×14, ⭐⭐⭐ ×12, ⭐⭐⭐⭐ ×4 (problems 21.23, 21.31, 21.37, 21.38). Estimated time: 5–8 hours. The full battery of convergence tests (comparison, ratio, root, integral, alternating) arrives in Chapter 22; the variable-term generalization to power and Taylor series is Chapter 23.