Chapter 24 — Self-Assessment Quiz

Q: Substituting into the Maclaurin series for and regrouping real and imaginary terms gives which formula, and what property of the series licenses the regrouping?

It gives Euler's formula . The regrouping is licensed by absolute convergence of the exponential series for all complex arguments, which permits rearranging terms without changing the sum (Sections 24.2–24.3; Chapter 22 on absolute vs. conditional convergence).

Q: Evaluate and state Euler's identity.

. Adding gives Euler's identity (Section 24.4).

Q: What are the values of at , and what does multiplication by that value do geometrically?

. Multiplying any complex number by rotates it a quarter turn () counterclockwise about the origin (Section 24.3).

Q: Write and purely in terms of complex exponentials. Why are these formulas the gateway to Fourier analysis?

and . They let us trade sines and cosines for clean exponentials when integrating, which is what makes the complex Fourier series work (Section 24.5).

Q: In polar form , what happens to the moduli and arguments when you multiply ?

: moduli multiply, arguments add. Multiplying complex numbers scales and rotates (Section 24.5).

Q: A calculator computes . Outline the two-step method most software math libraries use (Section 24.6).

Range reduction then a short series: first reduce the angle to a small interval near (e.g. ) using periodicity and symmetry, then sum a handful of Maclaurin (or minimax-polynomial) terms, stopping when the alternating-series error bound (Chapter 22) meets the tolerance (Section 24.6).

Q: State the Basel result and explain in one phrase where the comes from.

. The enters through the roots of (at multiples of ), which Euler used to factor and match the coefficient against its Taylor series (Section 24.8).

Q: Why is the Leibniz series "beautiful and almost useless," and what does Machin's formula fix (Section 24.9)?

It converges far too slowly — the error after terms is about , so two decimals need ~500 terms. Machin's formula evaluates at small arguments where the series races to convergence (Section 24.9).

Q: A Poisson random variable has probability generating function . Use it to find , and state which series recognition produced this PGF (Section 24.10).

, so . The PGF came from recognizing — the exponential series that opened the chapter (Section 24.10; term-by-term differentiation from Chapter 23).

DataField.Dev

Chapter 24 — Self-Assessment Quiz

10 questions covering Euler's formula and identity, complex exponentials, Fourier-series basics, series evaluation of functions, the Basel problem, and probability generating functions. Try each before opening the answer.

1. Substituting $x=i\theta$ into the Maclaurin series for $e^x$ and regrouping real and imaginary terms gives which formula, and what property of the series licenses the regrouping?

Answer

It gives **Euler's formula** $e^{i\theta}=\cos\theta+i\sin\theta$. The regrouping is licensed by **absolute convergence** of the exponential series for all complex arguments, which permits rearranging terms without changing the sum (Sections 24.2–24.3; Chapter 22 on absolute vs. conditional convergence).

2. Evaluate $e^{i\pi}$ and state Euler's identity.

Answer

$e^{i\pi}=\cos\pi+i\sin\pi=-1+0i=-1$. Adding $1$ gives **Euler's identity** $e^{i\pi}+1=0$ (Section 24.4).

3. What are the values of $e^{i\theta}$ at $\theta=\pi/2$, and what does multiplication by that value do geometrically?

Answer

$e^{i\pi/2}=\cos(\pi/2)+i\sin(\pi/2)=i$. Multiplying any complex number by $i$ **rotates it a quarter turn ($90°$) counterclockwise** about the origin (Section 24.3).

4. Write $\cos\theta$ and $\sin\theta$ purely in terms of complex exponentials. Why are these formulas the gateway to Fourier analysis?

Answer

$\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}$ and $\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$. They let us trade sines and cosines for clean exponentials when integrating, which is what makes the **complex Fourier series** $\sum c_n e^{inx}$ work (Section 24.5).

5. In polar form $z=re^{i\theta}$, what happens to the moduli and arguments when you multiply $z_1 z_2$?

Answer

$z_1 z_2 = r_1 r_2\,e^{i(\theta_1+\theta_2)}$: **moduli multiply, arguments add**. Multiplying complex numbers scales and rotates (Section 24.5).

6. A calculator computes $\sin(40°)$. Outline the two-step method most software math libraries use (Section 24.6).

Answer

**Range reduction then a short series**: first reduce the angle to a small interval near $0$ (e.g. $[-\pi/4,\pi/4]$) using periodicity and symmetry, then sum a handful of Maclaurin (or minimax-polynomial) terms, stopping when the alternating-series error bound (Chapter 22) meets the tolerance (Section 24.6).

7. The square wave equals $\dfrac{4}{\pi}\sum_{k=0}^{\infty}\dfrac{\sin((2k+1)x)}{2k+1}$. Why do only sine terms appear, and what is the Gibbs phenomenon?

Answer

The square wave is **odd**, so all cosine coefficients $a_n$ vanish and only sines survive. The **Gibbs phenomenon** is the persistent ~9% overshoot of the partial sums right beside each jump discontinuity; it does not vanish as $n\to\infty$, and at the jump itself the series converges to the **average** of the left/right values (Section 24.7).

8. State the Basel result and explain in one phrase where the $\pi$ comes from.

Answer

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. The $\pi$ enters through the **roots of $\sin x$** (at multiples of $\pi$), which Euler used to factor $\frac{\sin x}{x}$ and match the $x^2$ coefficient against its Taylor series (Section 24.8).

9. Why is the Leibniz series $\frac{\pi}{4}=1-\frac13+\frac15-\cdots$ "beautiful and almost useless," and what does Machin's formula fix (Section 24.9)?

Answer

It converges far too slowly — the error after $N$ terms is about $1/(2N)$, so two decimals need ~500 terms. **Machin's formula** $\frac{\pi}{4}=4\arctan\frac15-\arctan\frac1{239}$ evaluates $\arctan$ at small arguments where the series races to convergence (Section 24.9).

10. A Poisson random variable has probability generating function $G(s)=e^{\lambda(s-1)}$. Use it to find $E[X]$, and state which series recognition produced this PGF (Section 24.10).

Answer

$G'(s)=\lambda e^{\lambda(s-1)}$, so $E[X]=G'(1)=\lambda$. The PGF came from recognizing $\sum_n \frac{(\lambda s)^n}{n!}=e^{\lambda s}$ — the **exponential series** that opened the chapter (Section 24.10; term-by-term differentiation from Chapter 23).

Scoring Guide

Score	Interpretation
9–10	Excellent. You can derive Euler's formula and apply the chapter's tools fluently. Move on to Part V.
7–8	Solid. Review the one or two sections behind your misses, especially any derivation step.
5–6	Partial. Re-read Sections 24.3–24.5 (Euler's formula and consequences) and re-work the related exercises.
0–4	Revisit the chapter from Section 24.2, working each derivation by hand before re-testing.