Chapter 22 — Self-Assessment Quiz

Q: The -series converges if and only if: every

B) . The integral test compares it to , which converges exactly when . The borderline is the harmonic series, which diverges. §22.2.1.

Q: The integral test requires the function with to be: continuous only positive, continuous, and decreasing (at least eventually) any integrable function increasing and positive

B). All three conditions matter, but the one students forget is decreasing — without monotonicity the rectangle-sandwich argument collapses. §22.2 (Warning).

Q: For the ratio test, with , the series converges absolutely when: always

B) . If it diverges; if the test is inconclusive. The threshold matches the geometric series because the ratio test is local geometric comparison. §22.5.

Q: Which structure most cleanly signals the root test rather than the ratio test? a factorial the whole term raised to the -th power, terms decaying like alternating signs

B). When the entire term is an -th power, peels the exponent off in one line. Factorials point to the ratio test; decay points to limit comparison; alternating signs point to the alternating series test. §22.6.

Q: A series is conditionally convergent when: converges converges but diverges it converges to a positive number it diverges

B). It converges thanks to sign cancellation, yet falls apart once you take absolute values. If also converged, it would be absolutely convergent — the stronger condition. §22.8.

Q: The series (starting at ) converges; its exact sum is:

B) . The full series includes the term, which equals ; dropping it leaves . Convergence itself follows from the ratio test (). §22.5 (Worked Example 22.5.A); value derived in Chapter 23.

Q: The series is: divergent absolutely convergent conditionally convergent equal to

C) conditionally convergent. It converges by the alternating series test, but diverges. (Its sum is ; the closely related .) §22.7–22.8.

Q: The Riemann rearrangement theorem says that a conditionally convergent series can be rearranged to: any real sum you like, or to diverge only its original sum only nothing — its terms cannot be reordered

A). Reorder the same terms and you can hit any target, or make it diverge — infinite addition is not commutative. Absolutely convergent series, by contrast, are order-independent. §22.8 (Math Major Sidebar).

Q: For a convergent alternating series satisfying Leibniz's conditions, the error of the partial sum obeys: exactly no bound exists

B) . The error is at most the first omitted term, and it carries that term's sign. This certified bound is why alternating tails are prized in numerical computing. §22.7.

Q: Faced with a brand-new series , the *very first* thing to check is: the ratio test whether (the divergence test) a comparison series the integral test

B). If the series diverges immediately and you stop. Only if do you proceed down the framework. Remember: is necessary but not sufficient — the harmonic series has yet diverges. §22.10, Step 0.

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Chapter 22 — Self-Assessment Quiz

10 questions, ~20 minutes. Aim for 8/10. Each answer cites the section to revisit. This chapter rewards recognition over memorization — if you miss a question, re-run the series through the decision framework in §22.10.

1. 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) every $p$

Answer

**B) $p > 1$.** The integral test compares it to $\int_1^\infty x^{-p}\,dx$, which converges exactly when $p>1$. The borderline $p=1$ is the harmonic series, which diverges. *§22.2.1.*

2. The integral test requires the function $f$ with $a_n=f(n)$ to be:

A) continuous only
B) positive, continuous, and decreasing (at least eventually)
C) any integrable function
D) increasing and positive

Answer

**B).** All three conditions matter, but the one students forget is *decreasing* — without monotonicity the rectangle-sandwich argument collapses. *§22.2 (Warning).*

3. For the ratio test, with $\rho=\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$, the series converges absolutely when:

A) $\rho > 1$
B) $\rho < 1$
C) $\rho = 1$
D) always

Answer

**B) $\rho < 1$.** If $\rho>1$ it diverges; if $\rho=1$ the test is inconclusive. The threshold matches the geometric series because the ratio test *is* local geometric comparison. *§22.5.*

4. Which structure most cleanly signals the root test rather than the ratio test?

A) a factorial $n!$
B) the whole term raised to the $n$-th power, $a_n=(\,\cdots\,)^{n}$
C) terms decaying like $1/n^p$
D) alternating signs

Answer

**B).** When the entire term is an $n$-th power, $|a_n|^{1/n}$ peels the exponent off in one line. Factorials point to the ratio test; $1/n^p$ decay points to limit comparison; alternating signs point to the alternating series test. *§22.6.*

5. A series $\sum a_n$ is conditionally convergent when:

A) $\sum|a_n|$ converges
B) $\sum a_n$ converges but $\sum|a_n|$ diverges
C) it converges to a positive number
D) it diverges

Answer

**B).** It converges thanks to sign cancellation, yet falls apart once you take absolute values. If $\sum|a_n|$ also converged, it would be *absolutely* convergent — the stronger condition. *§22.8.*

6. The series $\displaystyle\sum_{n=1}^\infty \frac{1}{n!}$ (starting at $n=1$) converges; its exact sum is:

A) $e$
B) $e - 1$
C) $\pi$
D) $\ln 2$

Answer

**B) $e-1$.** The full series $\sum_{n=0}^\infty 1/n! = e$ includes the $n=0$ term, which equals $1$; dropping it leaves $e-1$. Convergence itself follows from the ratio test ($\rho=1/(n+1)\to 0$). *§22.5 (Worked Example 22.5.A); value derived in Chapter 23.*

7. The series $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n}}{n}$ is:

A) divergent
B) absolutely convergent
C) conditionally convergent
D) equal to $0$

Answer

**C) conditionally convergent.** It converges by the alternating series test, but $\sum 1/n$ diverges. (Its sum is $-\ln 2$; the closely related $\sum(-1)^{n+1}/n=+\ln 2$.) *§22.7–22.8.*

8. The Riemann rearrangement theorem says that a conditionally convergent series can be rearranged to:

A) any real sum you like, or to diverge
B) only its original sum
C) only $+\infty$
D) nothing — its terms cannot be reordered

Answer

**A).** Reorder the *same* terms and you can hit any target, or make it diverge — infinite addition is not commutative. Absolutely convergent series, by contrast, are order-independent. *§22.8 (Math Major Sidebar).*

9. For a convergent alternating series satisfying Leibniz's conditions, the error of the partial sum $S_N$ obeys:

A) $|L-S_N|\le b_N$
B) $|L-S_N|\le b_{N+1}$
C) $|L-S_N| = b_N$ exactly
D) no bound exists

Answer

**B) $|L-S_N|\le b_{N+1}$.** The error is at most the *first omitted term*, and it carries that term's sign. This certified bound is why alternating tails are prized in numerical computing. *§22.7.*

10. Faced with a brand-new series $\sum a_n$, the very first thing to check is:

A) the ratio test
B) whether $a_n\to 0$ (the divergence test)
C) a comparison series
D) the integral test

Answer

**B).** If $a_n\not\to 0$ the series diverges immediately and you stop. Only if $a_n\to 0$ do you proceed down the framework. Remember: $a_n\to 0$ is *necessary but not sufficient* — the harmonic series has $a_n\to 0$ yet diverges. *§22.10, Step 0.*

Scoring

9–10: Excellent. You are recognizing structure, not just executing tests — you are ready for power series in Chapter 23.
7–8: Solid. Re-read the decision framework (§22.10) and drill the "choose the test" problems (Exercises Part F).
5–6: Re-read §22.10 and Exercises Parts A–C; practice each test on several series until the routing is automatic.
Below 5: This is the hardest chapter in the book — that is expected. Slow down, work each test (§22.2–22.7) on five problems apiece, then retake the quiz.