Chapter 23 — Self-Assessment Quiz

10 questions, ~20 minutes. Aim for 8/10. Each answer cites the section to revisit.

1. The Maclaurin series for $e^x$ is:

  • A) $\sum_{n=0}^\infty x^n$
  • B) $\sum_{n=0}^\infty x^n/n!$
  • C) $\sum_{n=0}^\infty n!/x^n$
  • D) $\sum_{n=0}^\infty x^n/n$
Answer**B)** $\sum x^n/n!$, valid for all $x$. Every derivative of $e^x$ is $e^x$, so $f^{(n)}(0) = 1$ and $c_n = 1/n!$. *Section 23.4.*

2. What is the radius of convergence of $\sum_{n=0}^\infty x^n/n!$?

  • A) $1$ B) $\infty$ C) $0$ D) $2$
Answer**B) $\infty$.** Here $|c_n/c_{n+1}| = (n+1)! / n! = n+1 \to \infty$, so $R = \infty$ — the factorial in the denominator crushes every power of $x$. *Section 23.2 (Example 23.2.1).*

3. The Maclaurin series for $\sin x$ is:

  • A) $\sum (-1)^n x^{2n}/(2n)!$
  • B) $\sum (-1)^n x^{2n+1}/(2n+1)!$
  • C) $\sum x^n/n!$
  • D) None of the above
Answer**B)** $x - x^3/3! + x^5/5! - \cdots$ — only odd powers, because $\sin$ is an odd function. *Section 23.4.*

4. Using the Maclaurin series, $\sin(0.1)$ to four decimal places is:

  • A) $0.1000$ B) $0.0998$ C) $0.1010$ D) $0.0980$
Answer**B) $0.0998$.** $\sin(0.1) = 0.1 - \tfrac{(0.1)^3}{6} + \cdots = 0.1 - 0.0001\overline{6} + \cdots \approx 0.099833$. *Section 23.5 (error bounding).*

5. The Taylor polynomial $T_N$ centered at $a$ has error (Lagrange remainder):

  • A) $f^{(N)}(c)/N!$
  • B) $\dfrac{f^{(N+1)}(c)}{(N+1)!}(x-a)^{N+1}$ for some $c$ between $a$ and $x$
  • C) a constant independent of $N$
  • D) exactly zero
Answer**B)** the Lagrange remainder, with $c$ an unknown point between $a$ and $x$. Bounding $|f^{(N+1)}| \le M$ gives $|R_N| \le M|x-a|^{N+1}/(N+1)!$. *Section 23.5.*

6. The geometric series $\dfrac{1}{1-x} = \sum_{n=0}^\infty x^n$ converges for:

  • A) all $x$ B) $|x| < 1$ C) $x > 0$ D) $x = 0$ only
Answer**B) $|x| < 1$.** Radius $R = 1$; both endpoints $x = \pm 1$ diverge because the terms do not tend to $0$. *Section 23.2 (Example 23.2.2) and 23.4.*

7. The series $\ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} x^n/n$ has interval of convergence:

  • A) all $x$ B) $(-1, 1]$ C) $[0, \infty)$ D) $|x| < 2$
Answer**B) $(-1, 1]$.** At $x = -1$ the series becomes $-\sum 1/n$ (the divergent harmonic series); at $x = 1$ it is the alternating harmonic series, which converges to $\ln 2$ (Chapter 22). *Section 23.4 and 23.6.*

8. Integrating the series for $\dfrac{1}{1+x^2}$ term-by-term produces the series for:

  • A) $\arctan x$ B) $\ln(1+x)$ C) $e^x$ D) $\sin x$
Answer**A) $\arctan x$**, since $\tfrac{d}{dx}\arctan x = 1/(1+x^2)$. Setting $x=1$ gives the Leibniz formula $\pi/4 = 1 - \tfrac13 + \tfrac15 - \cdots$. *Section 23.6 (Worked Example 23.6.3).*

9. The series $\dfrac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^n x^{2n}$ has radius of convergence $R = 1$, even though $1/(1+x^2)$ is smooth on all of $\mathbb{R}$. Why?

  • A) The function actually blows up at $x = 1$.
  • B) The nearest singularities are the complex poles $\pm i$, at distance $1$ from the center.
  • C) The ratio test was applied incorrectly.
  • D) Smooth functions always have $R = 1$.
Answer**B)** The radius equals the distance from the center to the nearest singularity *in the complex plane*; $1/(1+z^2)$ has poles at $z = \pm i$, distance $1$ away. *Section 23.9.*

10. The integral $\int_0^a e^{-x^2}\,dx$ (the heart of the normal-curve anchor) is computed by:

  • A) finding its elementary antiderivative
  • B) expanding $e^{-x^2}$ as a Maclaurin series and integrating term-by-term
  • C) the chain rule
  • D) it cannot be computed at all
Answer**B)** No elementary antiderivative exists, so we substitute $-x^2$ into the $e^x$ series and integrate each power: $\int_0^a e^{-x^2}\,dx = a - \tfrac{a^3}{3} + \tfrac{a^5}{10} - \cdots$ This is the error function, and it closes the "area under the normal curve" anchor opened in Chapter 13. *Section 23.7.*

Scoring

  • 9–10: Excellent — you own the chapter. Move on to Chapter 24 (Euler's formula).
  • 7–8: Solid. Skim the sections flagged on any missed question.
  • 5–6: Re-read Section 23.4 (the seven standard series) and Section 23.5 (error bounds), then retry.
  • Below 5: Work through the index worked examples again, then derive at least five Taylor series by hand from Section 23.6 before retaking.