Chapter 20 — Self-Assessment Quiz

Ten questions, about 20 minutes. Each answer cites the section to review. Aim for 8/10. Try every question before opening an answer.

1. A sequence is a function whose domain is:

  • A) $\mathbb{R}$
  • B) $\mathbb{N}$
  • C) $[0, \infty)$
  • D) any nonempty set
Answer**B) $\mathbb{N}$.** A sequence assigns a real value $a_n$ to each natural number $n$ — it is a function sampled at the integers. *Section 20.1.*

2. $\displaystyle\lim_{n \to \infty} \frac{n^2 + 3n}{2n^2 - 5} = $

  • A) $0$
  • B) $\tfrac12$
  • C) $2$
  • D) $\infty$
Answer**B) $\tfrac12$.** Leading powers tie at $n^2$; divide top and bottom by $n^2$ to get $\dfrac{1 + 3/n}{2 - 5/n^2} \to \dfrac{1}{2}$ by the limit laws. *Section 20.4 (Worked Example 20.4.1).*

3. $\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = $

  • A) $1$
  • B) $e$
  • C) $\infty$
  • D) $2$
Answer**B) $e$.** This is the compound-interest standard limit $(1 + x/n)^n \to e^x$ with $x = 1$, and it *defines* Euler's number. *Sections 20.5 and 20.8.*

4. Which sequence diverges?

  • A) $a_n = 1/n$
  • B) $a_n = (-1)^n$
  • C) $a_n = n/(n+1)$
  • D) $a_n = \sin(\pi n)$
Answer**B) $(-1)^n$.** It oscillates between $-1$ and $+1$ forever and never settles near a single value, so it diverges — *bounded is not convergent.* (Note D: $\sin(\pi n) = 0$ for every integer $n$, so it converges to $0$.) *Section 20.3.*

5. The Squeeze Theorem lets you conclude $\dfrac{\sin n}{n} \to 0$ because:

  • A) $\sin n \to 0$
  • B) $-\tfrac1n \le \dfrac{\sin n}{n} \le \tfrac1n$ and both bounds $\to 0$
  • C) $\sin n$ is bounded, so the quotient is bounded
  • D) $1/n$ is monotone
Answer**B.** Since $-1 \le \sin n \le 1$, dividing by $n>0$ pins $\dfrac{\sin n}{n}$ between $\pm\tfrac1n$, both of which converge to $0$. (A is false: $\sin n$ does **not** converge.) *Section 20.4 (Worked Example 20.4.2).*

6. The Monotone Convergence Theorem states that a sequence converges if it is:

  • A) bounded
  • B) monotone
  • C) both monotone and bounded
  • D) Cauchy
Answer**C) both monotone and bounded.** A non-decreasing sequence bounded above (or non-increasing bounded below) converges — its power is that you need *not* know the limit in advance. *Section 20.6.*

7. In the growth hierarchy, which ordering is correct ($\prec$ means "is dominated by" as $n \to \infty$, with $p > 0$, $r > 1$)?

  • A) $\ln n \prec n^p \prec r^n \prec n! \prec n^n$
  • B) $n^n \prec n! \prec r^n \prec n^p \prec \ln n$
  • C) $\ln n \prec n^p \prec n! \prec r^n \prec n^n$
  • D) $r^n \prec n^p \prec n! \prec n^n \prec \ln n$
Answer**A.** Logarithms crawl, powers walk, exponentials run, factorials sprint, and $n^n$ outruns even factorials. This single ordering decides most sequence-limit questions. *Section 20.5.*

8. A fixed point of $f$ is a value $a^*$ with:

  • A) $f(a^*) = 0$
  • B) $f(a^*) = a^*$
  • C) $f'(a^*) = 0$
  • D) $a^* = 0$
Answer**B) $f(a^*) = a^*$.** A recursion $a_{n+1} = f(a_n)$ can only converge to a fixed point of $f$ — so finding limits of recursions reduces to solving $f(x) = x$. (A is a *root*, a different notion.) *Section 20.7.*

9. A fixed point $a^*$ of a differentiable $f$ is attracting (nearby iterations converge to it) when:

  • A) $f'(a^*) = 0$ exactly
  • B) $|f'(a^*)| < 1$
  • C) $f'(a^*) > 0$
  • D) $f$ is continuous at $a^*$
Answer**B) $|f'(a^*)| < 1$.** Near $a^*$ the error evolves as $e_{n+1} \approx f'(a^*)\,e_n$, so $|f'(a^*)| < 1$ shrinks the error geometrically. (Newton's method achieves $f'(a^*) = 0$, giving *quadratic* convergence — a special case.) *Section 20.7.*

10. A sequence converging to $0$ guarantees that the corresponding series converges.

  • A) True
  • B) False
Answer**B) False.** The terms $a_n = 1/n$ converge to $0$, yet the harmonic *series* $\sum 1/n$ diverges to infinity (Chapter 21). A sequence is a *list*; a series is a *sum*. Confusing "the terms go to zero" with "the sum is finite" is the single most common error in Part IV. *Section 20.9.*

Scoring

  • 9–10: Excellent — you have the foundation for series (Chapter 21) and convergence tests (Chapter 22).
  • 7–8: Solid. Revisit any missed standard limit in Section 20.5.
  • 5–6: Re-read Sections 20.4–20.6 (limit laws, squeeze, MCT) and redo the worked examples.
  • Below 5: Re-read the chapter from Section 20.3, focusing on the definition of convergence and the standard-limit catalog.