Chapter 22 — Exercises

40 problems on the integral test and $p$-series, direct and limit comparison, the ratio and root tests, alternating series and their error bound, absolute versus conditional convergence, and — above all — the strategy of choosing the right test. Difficulty is marked ⭐ (warm-up) through ⭐⭐⭐⭐ (challenge). Work the "choose the test" problems in Part F slowly: that recognition skill is what §22.10 is really teaching.

Part A — The Integral Test and $p$-Series (⭐ / ⭐⭐)

Use §22.2 throughout. Confirm positivity, continuity, and decreasing before invoking the integral test.

22.1 (⭐) State whether each $p$-series converges or diverges, citing the rule from §22.2.1: (a) $\sum 1/n^{3}$, (b) $\sum 1/n^{1/2}$, (c) $\sum 1/n^{1.001}$, (d) $\sum 1/n$.

22.2 (⭐) Use the integral test on $\sum_{n=1}^\infty 1/n^{3}$. Evaluate the improper integral $\int_1^\infty x^{-3}\,dx$ (Chapter 17) and state the verdict.

22.3 (⭐⭐) Apply the integral test to $\sum_{n=2}^\infty \dfrac{1}{n\ln n}$. (Substitute $u=\ln x$.) Compare your work to Worked Example 22.2.A.

22.4 (⭐⭐) Apply the integral test to $\sum_{n=2}^\infty \dfrac{1}{n(\ln n)^{2}}$. Why does this converge while 22.3 diverges? (See §22.2.2.)

22.5 (⭐⭐) Test $\sum_{n=1}^\infty \dfrac{n}{n^{2}+1}$ with the integral test. (Substitute $u=x^2+1$.)

22.6 (⭐⭐) Test $\sum_{n=1}^\infty n\,e^{-n^{2}}$ with the integral test. (Substitute $u=x^2$.)

Part B — Comparison and Limit Comparison (⭐⭐)

Use §22.3 and §22.4. For limit comparison, first find the leading-order decay, then pick $b_n=1/n^p$.

22.7 Direct comparison: show $\sum_{n=1}^\infty \dfrac{1}{n^{2}+n}$ converges by bounding it above by a convergent $p$-series.

22.8 Direct comparison: show $\sum_{n=1}^\infty \dfrac{1}{2^{n}+1}$ converges by bounding it above by a geometric series (Chapter 21).

22.9 Direct comparison: show $\sum_{n=3}^\infty \dfrac{\ln n}{n}$ diverges by bounding it below by the harmonic series. (Match the inequality direction to the conclusion — see the Common Pitfall in §22.3.)

22.10 Limit comparison: test $\sum_{n=1}^\infty \dfrac{3n+1}{n^{3}-2}$ against $1/n^2$.

22.11 Limit comparison: test $\sum_{n=1}^\infty \dfrac{1}{\sqrt{n^{2}+1}}$ against $1/n$.

22.12 Limit comparison: test $\sum_{n=1}^\infty \dfrac{n+4}{\sqrt{n^{3}+n}}$ against $n^{-1/2}$ (compare with Worked Example 22.4.C).

22.13 Limit comparison: test $\sum_{n=1}^\infty \sin\!\left(\dfrac1n\right)$. (Hint: $\sin x \approx x$ for small $x$, so compare with $1/n$.)

22.14 Limit comparison: test $\sum_{n=1}^\infty \left(1-\cos\dfrac1n\right)$. (Hint: $1-\cos x \approx x^{2}/2$, so compare with $1/n^{2}$.)

Part C — The Ratio and Root Tests (⭐⭐)

Use §22.5 (factorials, exponentials) and §22.6 ($n$-th powers).

22.15 Ratio test: $\sum_{n=1}^\infty \dfrac{n}{3^{n}}$.

22.16 Ratio test: $\sum_{n=1}^\infty \dfrac{n!}{n^{n}}$.

22.17 Ratio test: $\sum_{n=1}^\infty \dfrac{(n!)^{2}}{(2n)!}$.

22.18 Ratio test: $\sum_{n=1}^\infty \dfrac{3^{n}}{n!}$.

22.19 Root test: $\sum_{n=1}^\infty \left(\dfrac{n}{2n+1}\right)^{n}$.

22.20 Root test: $\sum_{n=2}^\infty \dfrac{1}{(\ln n)^{n}}$ (compare with Worked Example 22.6.B).

Part D — Alternating Series and the Error Bound (⭐⭐ / ⭐⭐⭐)

Use §22.7. The error bound $|L-S_N|\le b_{N+1}$ requires both Leibniz conditions.

22.21 (⭐⭐) Show $\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{\sqrt n}$ converges by the alternating series test.

22.22 (⭐⭐) Show $\sum_{n=2}^\infty \dfrac{(-1)^{n}}{\ln n}$ converges. Verify both Leibniz conditions explicitly.

22.23 (⭐⭐⭐) Approximate $\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^{3}}$ using the first $4$ terms. State a numerical error bound from §22.7, and give the sign of the error.

22.24 (⭐⭐⭐) For $\ln 2 = \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n}$, how many terms guarantee an error below $10^{-3}$? Compare your count to Worked Example 22.7.C and comment on the slow $1/N$ rate.

22.25 (⭐⭐⭐) For the Leibniz series $\dfrac{\pi}{4}=\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{2n-1}$, how many terms guarantee five-decimal accuracy ($|L-S_N|\le 5\times10^{-6}$)? Explain why this makes the Leibniz formula a terrible practical way to compute $\pi$.

22.26 (⭐⭐⭐) The series $\sum_{n=0}^\infty \dfrac{(-1)^{n}}{n!}=e^{-1}$ is alternating with $b_n=1/n!$. How many terms guarantee an error below $10^{-6}$? Contrast the term count with 22.24 — why is this one so much faster?

Part E — Absolute vs. Conditional Convergence (⭐⭐⭐)

Use §22.8. To classify, test $\sum|a_n|$ separately with the positive-term tools.

22.27 Classify $\sum_{n=1}^\infty \dfrac{(-1)^{n}}{n^{2}}$ (absolutely convergent, conditionally convergent, or divergent).

22.28 Classify $\sum_{n=1}^\infty \dfrac{(-1)^{n}}{\sqrt n}$.

22.29 Classify $\sum_{n=1}^\infty \dfrac{(-1)^{n}}{n^{3/2}}$.

22.30 Classify $\sum_{n=1}^\infty \dfrac{\cos n}{n^{2}}$. (Hint: $|\cos n|\le 1$; the signs are irregular, so the alternating series test does not apply — go straight to $\sum|a_n|$.)

22.31 Classify $\sum_{n=2}^\infty \dfrac{(-1)^{n}}{n\ln n}$. (Use 22.3 for the absolute series.)

22.32 Give your own example of (a) an absolutely convergent series and (b) a conditionally convergent series, and justify each in one line.

22.33 The rearrangement puzzle. The alternating harmonic series sums to $\ln 2\approx 0.693$ in natural order. Numerically compute partial sums of the rearrangement that takes one positive term, then two negative terms: $$1-\tfrac12-\tfrac14+\tfrac13-\tfrac16-\tfrac18+\tfrac15-\cdots$$ What value do the partial sums approach? Reconcile your answer with §22.8's Riemann rearrangement discussion. (Compute by hand to ~12 terms, or write a short loop and report the hand-checked result; do not rely on the machine to "discover" the value.)

Part F — Choose the Test (Strategy) (⭐⭐⭐ / ⭐⭐⭐⭐)

For each series, first name the test you would use and why (run §22.10's checklist in your head), then apply it.

22.34 (⭐⭐⭐) $\displaystyle\sum_{n=1}^\infty \frac{n^{10}}{2^{n}}$

22.35 (⭐⭐⭐) $\displaystyle\sum_{n=1}^\infty \frac{n^{2}+1}{n^{3}+5}$

22.36 (⭐⭐⭐) $\displaystyle\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^{n^{2}}$

22.37 (⭐⭐⭐⭐) Determine convergence of each, naming the deciding test: (a) $\displaystyle\sum_{n=2}^\infty \frac{(\ln n)^{2}}{n^{2}}$ (b) $\displaystyle\sum_{n=1}^\infty \frac{n!}{n^{2}\,2^{n}}$ (c) $\displaystyle\sum_{n=2}^\infty \left(\frac{\ln n}{n}\right)^{n}$ (d) $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n}\,n}{n^{2}+1}$ (also classify abs/cond)

22.38 (⭐⭐⭐⭐) One series, several tests. For $\sum_{n=1}^\infty \dfrac{1}{n^{2}+1}$, decide convergence by (a) direct comparison, (b) limit comparison, and (c) the integral test. Which is cleanest, and why does the ratio test fail here? (Recall §22.5, "When the Ratio Test Fails.")

Part G — Applied and Open-Ended (⭐⭐⭐⭐)

22.39 Statistical mechanics — the partition function. The Boltzmann partition function for a system with energy levels $E_n=n\varepsilon$ ($\varepsilon>0$) at temperature $T$ is $Z=\sum_{n=0}^\infty e^{-E_n/kT}=\sum_{n=0}^\infty \left(e^{-\varepsilon/kT}\right)^{n}$. Show this is geometric, identify the ratio $r$, and prove $Z$ converges for every $T>0$. Then write $Z$ in closed form and discuss the limit as $T\to\infty$.

22.40 (Open-ended.) Find a convergence question from your modeling-portfolio track (biology, economics, physics, or data science — see the portfolio callout in §22.11). State the series, identify which test from this chapter settles it, apply the test, and — if it converges — estimate how many terms you would need for two-decimal accuracy.

Total: 40 exercises. Estimated time: 8–12 hours. Worked solutions to odd-numbered problems are in appendices/answers-to-selected.md.