Chapter 20 — Exercises

Thirty-six problems on sequences, limits, the Squeeze Theorem, monotone convergence, recursion, and fixed points. Problems are tiered by difficulty: ⭐ warm-up, ⭐⭐ standard, ⭐⭐⭐ challenging, ⭐⭐⭐⭐ proof or synthesis. Parts A–H build from mechanical limits to a full ε-N proof. Answers to odd-numbered problems appear in the back-of-book answer key.

Total: 36 problems. Estimated time: 6–9 hours.

Part A — Warm-Up: Terms and Behavior (⭐)

20.1 Write the first five terms of $a_n = \dfrac{(-1)^{n+1}}{n^2}$. Does the sequence appear to converge, and if so, to what?

20.2 Write the first five terms of the recursive sequence $a_1 = 4$, $a_{n+1} = \tfrac12 a_n + 1$. Guess the limit.

20.3 Classify each sequence as convergent or divergent (no proof needed): (a) $a_n = 5$, (b) $a_n = (-1)^n n$, (c) $a_n = \dfrac{n}{n+1}$, (d) $a_n = \cos(\pi n)$.

20.4 For $a_n = \dfrac{2n-1}{n}$, find an index $N$ beyond which every term lies within $0.01$ of its limit $L = 2$.

20.5 Is $a_n = \dfrac{n}{n+1}$ monotone? Is it bounded above? Below?

20.6 Using the growth hierarchy of §20.5, state (without computation) which of $n^{10}$ and $2^n$ eventually dominates, and which of $n!$ and $3^n$ eventually dominates.

Part B — Compute Limits with Limit Laws (⭐⭐)

For each, find $\lim_{n \to \infty} a_n$ or state divergence. Cite the limit law or standard limit you use (§20.4–20.5).

20.7 $a_n = \dfrac{3n + 1}{2n - 5}$

20.8 $a_n = \dfrac{n}{n^2 + 1}$

20.9 $a_n = \dfrac{n^2 + 1}{n^3 - 1}$

20.10 $a_n = \dfrac{2^n + 3^n}{3^n}$

20.11 $a_n = \left(1 + \dfrac{1}{n}\right)^{2n}$ (Hint: relate to the standard limit $(1+x/n)^n \to e^x$ of §20.5.)

20.12 $a_n = \dfrac{\ln n}{n}$

20.13 $a_n = n^{1/n}$

20.14 $a_n = \sqrt{n+1} - \sqrt{n}$ (Hint: rationalize.)

Part C — The Squeeze Theorem (⭐⭐)

20.15 Find $\lim_{n\to\infty} \dfrac{\cos n}{n}$ by squeezing between two sequences that both go to $0$. State your bounds.

20.16 Find $\lim_{n\to\infty} \dfrac{(-1)^n}{n}$ and explain why the Squeeze Theorem (not the limit laws alone) is the clean tool here.

20.17 Show $\lim_{n\to\infty} \dfrac{n + \sin n}{n^2} = 0$. Identify the squeezing bounds.

20.18 Find $\lim_{n\to\infty} \dfrac{2^n}{n!}$ by showing that for $n \ge 2$, $\;0 < \dfrac{2^n}{n!} \le \dfrac{4}{n}$, then squeezing. (Hint: write $\dfrac{2^n}{n!} = \dfrac{2}{1}\cdot\dfrac{2}{2}\cdot\dfrac{2}{3}\cdots\dfrac{2}{n}$; the first two factors multiply to $2$, every factor from $3$ to $n-1$ is $\le 1$, and the last factor is $\tfrac2n$, giving the bound $2\cdot\tfrac2n = \tfrac4n$.)

Part D — Monotone Convergence (⭐⭐⭐)

20.19 Show $a_n = \dfrac{n}{n+1}$ is increasing and bounded above by $1$, so it converges by the Monotone Convergence Theorem (§20.6). Then find the limit and confirm.

20.20 Define $a_1 = 1$, $a_{n+1} = \tfrac12\!\left(a_n + \dfrac{3}{a_n}\right)$ (the Babylonian method for $\sqrt 3$). (a) Show $a_{n+1} \ge \sqrt 3$ for $n \ge 1$ using AM–GM. (b) Show the sequence is non-increasing for $n \ge 2$. (c) Apply the MCT and solve the fixed-point equation to conclude $a_n \to \sqrt 3$.

20.21 Define $a_1 = 1$, $a_{n+1} = \sqrt{2 + a_n}$. (a) Show by induction that $a_n < 2$ for all $n$. (b) Show the sequence is increasing. (c) Conclude it converges and find the limit by solving $L = \sqrt{2 + L}$.

20.22 Let $a_n = 1 + \tfrac12 + \tfrac13 + \cdots + \tfrac1n - \ln n$. Show $\{a_n\}$ is decreasing and bounded below by $0$, hence convergent. (Its limit is the Euler–Mascheroni constant $\gamma \approx 0.5772$; see Further Reading.) Caution: this is the sequence $a_n$, not the harmonic series $\sum 1/n$, which diverges (Chapter 21).

Part E — Recursive Sequences and Fixed Points (⭐⭐⭐)

20.23 For $a_{n+1} = \cos(a_n)$, $a_0 = 1$ (radians), identify the limit as the fixed point of $\cos$ and verify the stability criterion $|f'(a^*)| < 1$ holds (§20.7). State whether the approach is monotone or oscillating, and why.

20.24 Find the fixed point(s) of $f(x) = \tfrac12 x + 1$ and show the recursion $a_{n+1} = f(a_n)$ converges to it from any start by checking $|f'| < 1$.

20.25 The iteration $x_{n+1} = 2x_n(1 - x_n)$ (logistic map at $r = 2$) has fixed points where $x = 2x(1-x)$. (a) Find both fixed points. (b) Use $|f'(x^*)| < 1$ to determine which is attracting. (c) Starting from $x_0 = 0.1$, argue the iteration converges to the attracting fixed point.

20.26 (Newton's method as a recursive sequence.) Newton's iteration for $g(x) = x^2 - 5$ is $a_{n+1} = a_n - \dfrac{a_n^2 - 5}{2a_n} = \tfrac12\!\left(a_n + \dfrac{5}{a_n}\right)$. (a) Show its fixed points are $\pm\sqrt 5$. (b) Compute $f'(x)$ for $f(x) = \tfrac12(x + 5/x)$ and verify $f'(\sqrt 5) = 0$, explaining the quadratic convergence of §20.7. (c) In Python, iterate from $a_1 = 2$ and report the error $|a_n - \sqrt 5|$ at each step.

Part F — Applications (⭐⭐⭐)

20.27 (Economics — compound interest.) A savings account earns 5% annual interest, compounded daily. Compute the effective annual yield $(1 + 0.05/365)^{365} - 1$ and compare it to the continuously compounded limit $e^{0.05} - 1$ (§20.8). How close are they?

20.28 (Medicine — drug accumulation.) A patient takes a dose $D$ at intervals of one half-life, so the blood concentration just after the $n$-th dose follows $C_n = \tfrac12 C_{n-1} + D$, with $C_0 = D$. (a) Find the fixed point (steady-state concentration) $C^*$. (b) Show $\{C_n\}$ is increasing and bounded above by $C^*$, hence converges to it. (c) After how many doses does $C_n$ first exceed $0.95\,C^*$?

20.29 (Biology — Fibonacci and the golden ratio.) For Fibonacci $F_1 = F_2 = 1$, $F_{n+1} = F_n + F_{n-1}$, let $r_n = F_{n+1}/F_n$. (a) Show $r_n$ satisfies $r_n = 1 + 1/r_{n-1}$. (b) Assuming $r_n \to L$, find $L$ by solving the fixed-point equation $L = 1 + 1/L$ and conclude $L = \phi = (1+\sqrt5)/2$ (§20.7).

20.30 (Ecology — sustainable harvest.) A population satisfies $P_{n+1} = P_n(1 + r) - h$, where $r$ is the growth rate and $h$ the annual harvest. (a) Find the equilibrium $P^*$ making $P_n$ constant. (b) For $r = 0.1$, what harvest $h$ holds the population at $P^* = 5000$? (c) Is the equilibrium stable? (Check $|f'(P^*)|$ for $f(P) = P(1+r) - h$.)

20.31 (Data science — fixed-point view of gradient descent.) For the update $\theta_{n+1} = \theta_n - \eta\,(2\theta_n)$ minimizing $L(\theta) = \theta^2$, write it as $\theta_{n+1} = f(\theta_n)$. (a) Identify $f$ and its fixed point. (b) Show convergence requires $|1 - 2\eta| < 1$, i.e. $0 < \eta < 1$, connecting the learning rate directly to the stability criterion of §20.7.

Part G — sympy and Visualization (⭐⭐)

For these, write the code and hand-predict the output before reasoning about it; do not rely on execution to learn the answer.

20.32 Use sp.limit(a_n, n, sp.oo) to verify your answers to 20.7, 20.11, and 20.13. Write the expected output as a comment for each.

20.33 Plot the first 30 terms of $a_n = (1 + 1/n)^n$ against $n$ with a horizontal line at $y = e$. Describe in one sentence how the dots approach the line (from above or below, fast or slow).

20.34 Implement the Babylonian iteration for $\sqrt 7$ and tabulate $|a_n - \sqrt 7|$ for $n = 1,\ldots,6$. Predict by hand that the error roughly squares each step (quadratic convergence, §20.7) and confirm the table matches.

Part H — Proof and Synthesis (⭐⭐⭐⭐)

20.35 (ε-N proof.) Using only the definition of §20.3, prove that $\displaystyle\lim_{n\to\infty} \frac{2n+1}{n+3} = 2$. That is: given $\varepsilon > 0$, produce an explicit $N$ (in terms of $\varepsilon$) such that $n \ge N \implies \left|\dfrac{2n+1}{n+3} - 2\right| < \varepsilon$. (Hint: simplify the absolute value to $\dfrac{5}{n+3}$ first.)

20.36 (Synthesis — MCT versus knowing the limit.) Explain in a short paragraph why the Monotone Convergence Theorem is so powerful for recursive sequences: it certifies convergence before you know the limit. Then give one concrete example (your own, or 20.21) where you prove convergence by MCT and only afterward solve the fixed-point equation $L = f(L)$ to identify the limit — and explain why doing those steps in the reverse order would be a logical error (§20.7 Common Pitfall).