Chapter 21 — Self-Assessment Quiz

Q: An infinite series is *defined* to be: the result of adding all the at once , the limit of the partial sums the limit of the terms the largest partial sum

B. A series is the limit of its sequence of partial sums — not a single act of addition (and not the limit of the terms). Convergence of the series IS convergence of . Section 21.2.

Q: A geometric series (with ) converges if and only if: B) C) D)

B) . Then and the partial sums settle at . For the powers of do not die out, so the series diverges. Section 21.3.

Q: B) C) D)

B) 2. Geometric with , : sum . (Starting at the first term is .) Section 21.3.

Q: The harmonic series : converges to B) diverges C) equals D) converges to

B) Diverges. By Oresme's grouping argument the partial sums exceed and grow without bound (like ) — even though the terms . Section 21.5.

Q: The divergence (-th term) test says: if , then : converges B) diverges C) is inconclusive D) equals

B) Diverges. This is the contrapositive of "if converges then ." Section 21.6.

Q: True or false: " implies converges." True B) False C) True only for positive D) True only for decreasing

B) False. This is the single most important caveat in the chapter. The harmonic series has yet diverges. The divergence test is a one-way tool: it can prove divergence but never convergence. Section 21.6 (Warning).

Q: B) C) D) diverges

B) 1. Partial fractions give ; the sum telescopes to . Section 21.4.

Q: Which rational number equals the repeating decimal ? B) C) exactly D) it is irrational

B) . Write , geometric with , : sum . Section 21.3.

Q: A perpetuity pays \ per year forever; the annual discount rate is . Its present value is: B) C) D)

B) . is geometric with first term and ratio , summing to . Sections 21.3 and 21.10.

Q: The -series converges if and only if: B) C) D) all

B) . The harmonic case is the borderline failure; gives the convergent Basel sum . (Proved with the integral test in Chapter 22.) Section 21.8.

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Chapter 21 — Self-Assessment Quiz

10 questions, ~25 minutes. Work each by hand before opening the answer. Each answer cites the section to revisit.

1. An infinite series $\sum_{n=1}^\infty a_n$ is defined to be: - A) the result of adding all the $a_n$ at once - B) $\displaystyle\lim_{N \to \infty} S_N$, the limit of the partial sums $S_N = a_1 + \cdots + a_N$ - C) the limit of the terms $a_n$ - D) the largest partial sum

Answer

**B.** A series is the limit of its sequence of partial sums — not a single act of addition (and *not* the limit of the terms). Convergence of the series IS convergence of $\{S_N\}$. *Section 21.2.*

2. A geometric series $\displaystyle\sum_{n=0}^\infty a r^n$ (with $a \neq 0$) converges if and only if: - A) $r > 0$ B) $|r| < 1$ C) $r < 1$ D) $r \neq 1$

Answer

**B) $|r| < 1$.** Then $r^{N+1} \to 0$ and the partial sums settle at $a/(1-r)$. For $|r| \ge 1$ the powers of $r$ do not die out, so the series diverges. *Section 21.3.*

3. $\displaystyle\sum_{n=0}^\infty \left(\tfrac{1}{2}\right)^n = $ - A) $1$ B) $2$ C) $\infty$ D) $\tfrac12$

Answer

**B) 2.** Geometric with $a = 1$, $r = \tfrac12$: sum $= \dfrac{1}{1 - 1/2} = 2$. (Starting at $n=0$ the first term is $r^0 = 1$.) *Section 21.3.*

4. The harmonic series $\displaystyle\sum_{n=1}^\infty \tfrac1n$: - A) converges to $1$ B) diverges C) equals $\pi^2/6$ D) converges to $\ln 2$

Answer

**B) Diverges.** By Oresme's grouping argument the partial sums exceed $1 + \tfrac{k}{2}$ and grow without bound (like $\ln N$) — even though the terms $\tfrac1n \to 0$. *Section 21.5.*

5. The divergence ($n$-th term) test says: if $\displaystyle\lim_{n\to\infty} a_n \neq 0$, then $\sum a_n$: - A) converges B) diverges C) is inconclusive D) equals $0$

Answer

**B) Diverges.** This is the contrapositive of "if $\sum a_n$ converges then $a_n \to 0$." *Section 21.6.*

6. True or false: "$a_n \to 0$ implies $\sum a_n$ converges." - A) True B) False C) True only for positive $a_n$ D) True only for decreasing $a_n$

Answer

**B) False.** This is the single most important caveat in the chapter. The harmonic series $\sum \tfrac1n$ has $a_n \to 0$ yet diverges. The divergence test is a one-way tool: it can prove divergence but never convergence. *Section 21.6 (Warning).*

7. $\displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)} = $ - A) $0$ B) $1$ C) $\pi/2$ D) diverges

Answer

**B) 1.** Partial fractions give $\tfrac1n - \tfrac{1}{n+1}$; the sum telescopes to $S_N = 1 - \tfrac{1}{N+1} \to 1$. *Section 21.4.*

8. Which rational number equals the repeating decimal $0.\overline{3}$? - A) $\tfrac{3}{10}$ B) $\tfrac13$ C) $0.3$ exactly D) it is irrational

Answer

**B) $\tfrac13$.** Write $0.\overline{3} = \sum_{n=1}^\infty \tfrac{3}{10^n}$, geometric with $a = \tfrac{3}{10}$, $r = \tfrac{1}{10}$: sum $= \dfrac{3/10}{9/10} = \tfrac13$. *Section 21.3.*

9. A perpetuity pays \$C$ per year forever; the annual discount rate is $r$. Its present value is: - A) $C \cdot r$ B) $C/r$ C) $r/C$ D) $C/(1-r)$

Answer

**B) $C/r$.** $PV = \sum_{n=1}^\infty \dfrac{C}{(1+r)^n}$ is geometric with first term $\tfrac{C}{1+r}$ and ratio $\tfrac{1}{1+r}$, summing to $C/r$. *Sections 21.3 and 21.10.*

10. The $p$-series $\displaystyle\sum_{n=1}^\infty \frac{1}{n^p}$ converges if and only if: - A) $p > 0$ B) $p > 1$ C) $p < 1$ D) all $p$

Answer

**B) $p > 1$.** The harmonic case $p = 1$ is the borderline failure; $p = 2$ gives the convergent Basel sum $\pi^2/6$. (Proved with the integral test in Chapter 22.) *Section 21.8.*

Scoring Guide

9–10 correct — Excellent. You own the geometric-series formula, the harmonic caveat, and the divergence test. You are ready for the convergence tests of Chapter 22.
7–8 correct — Solid. Re-read the section flagged on each missed answer; pay special attention to Q6 (the $a_n \to 0$ caveat).
5–6 correct — Developing. Re-study §21.3 (geometric series) and §21.8 (p-series), then re-attempt the geometric and telescoping exercises.
Below 5 — Slow down. These results are foundational for the rest of Part IV. Re-read the chapter, redo the Check Your Understanding boxes, and try Parts A–D of the exercises before retaking.