Chapter 23 — Exercises

38 problems on power series, radius and interval of convergence, Taylor and Maclaurin series, error bounds, term-by-term operations, and applications.

Work these with paper, pencil, and the seven memorized series of Section 23.4. A calculator is for checking, not for finding — the whole point of the chapter is that you can produce these numbers yourself, with a certified error bound, from an infinite polynomial.

Difficulty tiers: ⭐ routine (one definition, one step) · ⭐⭐ standard (multi-step, the daily bread of the chapter) · ⭐⭐⭐ challenging (combines several ideas) · ⭐⭐⭐⭐ deep (proof, modeling, or open-ended).

Part A — Radius and Interval of Convergence (⭐)

Use the ratio-test formula $R = \lim_{n\to\infty}\left|c_n/c_{n+1}\right|$ from Section 23.2, then test the endpoints by hand with the convergence tools of Chapter 22.

A1. Find the radius of convergence of $\displaystyle\sum_{n=0}^\infty \frac{x^n}{3^n}$.

A2. Find the radius of convergence of $\displaystyle\sum_{n=0}^\infty \frac{x^n}{n!}$.

A3. Find the radius of convergence of $\displaystyle\sum_{n=0}^\infty n!\,x^n$.

A4. Find the radius of convergence of $\displaystyle\sum_{n=1}^\infty \frac{x^n}{n^2}$.

A5. For the series in A4, determine the full interval of convergence (test both endpoints).

A6. State the interval of convergence of the geometric series $\displaystyle\sum_{n=0}^\infty x^n$, including endpoint behavior.

A7. True or false: if a power series centered at $a = 2$ has $R = 4$, it converges on $(-2, 6)$ and possibly at the endpoints. Justify in one sentence.

A8. Why does the ratio test fail to decide convergence exactly at $|x - a| = R$? Answer in one sentence, referencing Section 23.2.

Part B — Building Maclaurin Series by Substitution (⭐⭐)

Use the reference table in Section 23.4 and the substitution rule of Section 23.6. Do not compute derivatives.

B1. Write the Maclaurin series for $\sin(3x)$ (general term).

B2. Write the first three nonzero terms of the Maclaurin series for $e^{-2x}$.

B3. Write the Maclaurin series for $\dfrac{1}{1 + x^3}$ (general term) and state its radius of convergence.

B4. Write the first three nonzero terms of the Maclaurin series for $\cos(x^2)$.

B5. Write the Maclaurin series for $x\,e^{x}$ (general term).

B6. Using the binomial series of Section 23.4, write the first three terms of $\sqrt{1 + x}$.

Part C — Taylor and Maclaurin Series from the Definition (⭐⭐ / ⭐⭐⭐)

Use the forced coefficient formula $c_n = f^{(n)}(a)/n!$ directly (Section 23.4).

C1. (⭐⭐) Compute the Maclaurin series of $f(x) = \cos x$ from the derivative formula, through the $x^4$ term, confirming the table.

C2. (⭐⭐) Find the Taylor series of $f(x) = e^x$ centered at $a = 1$. (Hint: $e^x = e\cdot e^{\,x-1}$, then expand $e^{x-1}$.)

C3. (⭐⭐) Find the Taylor polynomial $T_2(x)$ of $f(x) = \sqrt{x}$ centered at $a = 4$.

C4. (⭐⭐) Find the Taylor polynomial $T_3(x)$ of $f(x) = \ln x$ centered at $a = 1$, and identify the general-term pattern.

C5. (⭐⭐⭐) Find the Maclaurin series of $f(x) = \dfrac{1}{(1-x)^2}$ two ways: (a) by differentiating the geometric series term-by-term (Section 23.6), and (b) by squaring the geometric series via the Cauchy product (Section 23.6). Confirm they agree through $x^3$.

C6. (⭐⭐⭐) Find the Taylor series of $\sin x$ centered at $a = \pi/2$. What familiar Maclaurin series do you recognize, and why?

Part D — Taylor's Theorem and Error Bounds (⭐⭐ / ⭐⭐⭐)

Use the Lagrange remainder bound $|R_N(x)| \le \dfrac{M}{(N+1)!}\,|x-a|^{N+1}$ from Section 23.5, where $M$ bounds $|f^{(N+1)}|$ between $a$ and $x$.

D1. (⭐⭐) Approximate $e^{0.2}$ with the degree-3 Maclaurin polynomial, then bound the error using $M = e^{0.2} < 1.25$.

D2. (⭐⭐) Approximate $\cos(0.3)$ with the terms through $x^2$, then bound the error.

D3. (⭐⭐) Approximate $\ln(1.1)$ using the first two nonzero terms of the $\ln(1+x)$ series. Since the series alternates, bound the error by the first omitted term (the alternating series estimate of Chapter 22).

D4. (⭐⭐) How many terms of the $\sin x$ series guarantee $\sin(0.5)$ to within $10^{-6}$? Use $M = 1$.

D5. (⭐⭐⭐) You want $e$ (that is, $e^1$) accurate to within $10^{-5}$ from the Maclaurin series. Find the smallest $N$, bounding the relevant derivative by $f^{(N+1)}(c) = e^c \le e < 3$ on $[0,1]$.

D6. (⭐⭐⭐) For $f(x) = \cos x$ approximated at $x = 1$ by its degree-$N$ Maclaurin polynomial, find the smallest even $N$ giving error below $10^{-4}$.

Part E — Term-by-Term Differentiation and Integration (⭐⭐⭐)

All four problems use the term-by-term license of Sections 23.3 and 23.6.

E1. Starting from $\dfrac{1}{1-x} = \sum x^n$, derive the series for $\dfrac{1}{(1-x)^3}$ by differentiating twice. State the radius.

E2. Derive the Maclaurin series of $\arctan x$ by substituting $-x^2$ into the geometric series and integrating (Section 23.6). Then set $x = 1$ to write the Leibniz formula for $\pi/4$, and explain in one sentence why it converges slowly.

E3. Find a power series for the sine-integral $\displaystyle\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt$ by integrating the series for $\sin t / t$ term-by-term.

E4. Use series arithmetic to evaluate $\displaystyle\lim_{x\to 0}\frac{1 - \cos x}{x^2}$ without L'Hôpital (Section 23.6).

Part F — Approximating the "Impossible" Integral (⭐⭐⭐ / ⭐⭐⭐⭐)

This is the chapter's anchor payoff (Section 23.7): integrals with no elementary antiderivative, computed by integrating a Taylor series.

F1. (⭐⭐⭐) Write the Maclaurin series for $e^{-x^2}$ (general term, Section 23.7).

F2. (⭐⭐⭐) Integrate term-by-term to obtain a series for $\displaystyle\int_0^a e^{-x^2}\,dx$.

F3. (⭐⭐⭐) Use the first four terms of F2 to approximate $\displaystyle\int_0^{1/2} e^{-x^2}\,dx$. Because the series alternates, bound the error by the first omitted term.

F4. (⭐⭐⭐) Approximate $\displaystyle\int_0^1 \frac{1 - \cos x}{x^2}\,dx$ by expanding the integrand as a series first, then integrating.

F5. (⭐⭐⭐⭐) Using the Computational Note in Section 23.7, explain why the term-by-term series for $\int_0^a e^{-x^2}\,dx$ becomes numerically unreliable for large $a$, even though it converges for every $a$. What do production libraries switch to instead?

Part G — Applications and Proof (⭐⭐⭐⭐)

G1. (⭐⭐⭐⭐, physics) Starting from the relativistic kinetic energy $E_k = (\gamma - 1)mc^2$ with $\gamma = (1 - v^2/c^2)^{-1/2}$, use the binomial series (Section 23.8) to show $E_k = \tfrac12 mv^2 + \tfrac38 \dfrac{mv^4}{c^2} + \cdots$. At $v = 0.1c$, what fraction of the Newtonian term is the first correction?

G2. (⭐⭐⭐⭐, statistics) The standard-normal CDF satisfies $\Phi(z) = \tfrac12 + \tfrac{1}{\sqrt{2\pi}}\displaystyle\sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{2^n\,n!\,(2n+1)}$ (Section 23.7). Use the first three terms to estimate $\Phi(0.5) - 0.5$, the probability that a standard normal lands in $[0, 0.5]$, and compare to the tabulated $0.1915$.

G3. (⭐⭐⭐⭐, proof) Prove that the Maclaurin series of $\cos x$ converges to $\cos x$ for every real $x$, by showing the Lagrange remainder $R_N(x) \to 0$ as $N \to \infty$ (Section 23.10). Where exactly do you use that all derivatives of $\cos$ are bounded by 1?

Answers to Selected Exercises

Brief answers appear in appendices/answers-to-selected.md; full solutions live in the Instructor Guide. A few checkpoints:

• A1: $R = 3$; interval $(-3,3)$ — both endpoints diverge since the terms $(\pm1)^n$ do not tend to $0$.
• A5: Interval $[-1, 1]$; at both endpoints the series compares to $\sum 1/n^2$, which converges (Chapter 22).
• B1: $\sin(3x) = \displaystyle\sum_{n=0}^\infty \frac{(-1)^n (3x)^{2n+1}}{(2n+1)!}$.
• B6: $\sqrt{1+x} = 1 + \tfrac12 x - \tfrac18 x^2 + \cdots$.
• C4: $T_3(x) = (x-1) - \tfrac12(x-1)^2 + \tfrac13(x-1)^3$; general term $\dfrac{(-1)^{n+1}}{n}(x-1)^n$.
• D1: $T_3(0.2) = 1 + 0.2 + 0.02 + 0.001\overline{3} = 1.221\overline{3}$; bound $|R_3| \le \tfrac{1.25}{24}(0.2)^4 \approx 8.3\times10^{-5}$.
• D4: Need $\tfrac{1}{(N+1)!}(0.5)^{N+1} < 10^{-6}$; $N = 5$ works (terms through $x^5/5!$).
• E4: $\dfrac{1 - \cos x}{x^2} = \dfrac{1}{2} - \dfrac{x^2}{24} + \cdots \to \tfrac12$.
• F3: $\tfrac12 - \tfrac{(1/2)^3}{3} + \tfrac{(1/2)^5}{10} - \tfrac{(1/2)^7}{42} \approx 0.461281$; error $< \tfrac{1}{9\cdot 2^9} \approx 2.2\times10^{-4}$.
• G1: First correction $/$ Newtonian $= \tfrac34\,\dfrac{v^2}{c^2} = 0.0075$, i.e. $0.75\%$ at $v = 0.1c$.
• G2: $\tfrac{1}{\sqrt{2\pi}}\left(0.5 - \tfrac{0.125}{6} + \tfrac{0.03125}{40}\right) \approx 0.3989 \times 0.4804 \approx 0.1916$, matching the table.