Chapter 3 — The Convergence-Test Toolkit

Why this chapter matters: this is the heart of BC series. Roughly 80% of AP convergence problems boil down to "pick the right test from this menu." Practice until you can identify the right test from a one-second glance at the general term.

By the end of this chapter you will:

1. Master 7 major tests (n-th term, integral, p-series, comparison, limit comparison, ratio, root)
2. Use a decision tree to pick a test
3. Handle every standard AP convergence question

3.1 The Decision Tree — Spirit of the Chapter

Given a series $\sum a_n$, ask these questions in order:

Look at a_n
│
├─ lim a_n ≠ 0?              → Divergent (n-th term test, §3.2)
│
├─ Form a·r^n?               → Geometric series (Ch 2)
│
├─ Form $\frac{1}{n}$^p?               → p-series (Ch 2)
│
├─ Has (-1)^n alternation?   → Alternating Series Test (Ch 4)
│
├─ Has n!, r^n, or n^n?      → Ratio test or Root test (§3.6, 3.7)
│
├─ a_n = f(n) with f         → Integral test (§3.3)
│   positive & decreasing?
│
├─ Looks like a known        → Comparison or Limit Comparison (§3.4, 3.5)
│   conv/div series?
│
└─ None of the above?        → Try telescoping (Ch 2.5)

🔑 Meta-rule: scan the shape of $a_n$. Factorials/exponentials → ratio/root. Rational expressions → comparison/limit comparison. Logs/trig → integral test.

🎵 Audio 1 — Decision-tree overview — audio/ch3_01_decision_tree.mp3

3.2 n-th Term Test (review)

If $\lim_{n\to\infty} a_n \ne 0$, then $\sum a_n$ diverges. The converse is false.

Already covered in §2.4. Use as your first filter.

3.3 Integral Test

Hypotheses (all three required):

• $a_n = f(n)$ where $f$ is positive, continuous, and monotonically decreasing on $[1, \infty)$

Conclusion:
$$\sum_{n=1}^{\infty} a_n \text{ and } \int_1^{\infty} f(x)\,dx \text{ converge or diverge together}$$

Intuition: integrals are the continuous analog of sums. Picture rectangles of width 1 and height $a_n$; monotone decrease ensures the rectangles' total area is sandwiched between $\int_1^N f$ and $\int_0^{N-1} f$.

🎵 Audio 2 — Integral test — audio/ch3_02_integral.mp3

Example

Decide $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n \ln n}$.

Step 1 — Verify hypotheses: $f(x) = \frac{1}{x \ln x}$ is positive, continuous, decreasing on $[2, \infty)$. ✓

Step 2 — Evaluate the integral:
$$\int_2^{\infty} \frac{1}{x \ln x} dx \xrightarrow{u = \ln x} \int_{\ln 2}^{\infty} \frac{du}{u} = \infty$$

Integral diverges → series diverges.

🔑 Strategy: when $\ln n$ appears in the denominator, substitute $u = \ln x$.

⚠️ Trap: the integral test cannot give you the value of the sum. Only convergence/divergence. (For example, $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ but $\int_1^\infty x^{-2}\,dx = 1$.)

3.4 Direct Comparison Test (DCT)

Hypotheses: $0 \le a_n \le b_n$ for all sufficiently large $n$ (i.e. positive-term series).

Conclusion:

• $\sum b_n$ converges $\Rightarrow \sum a_n$ converges
• $\sum a_n$ diverges $\Rightarrow \sum b_n$ diverges

Analogy: if your wallet contains less than a billionaire's, and the billionaire is finite, then so is yours. The reverse argument is for divergence.

🎵 Audio 3 — Direct comparison — audio/ch3_03_compare.mp3

Example

Decide $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 + n}$.

Reasoning: $\frac{1}{n^2 + n} \le \frac{1}{n^2}$, and $\sum \frac{1}{n^2}$ converges (p-series, $p=2$). DCT → converges.

🔑 Strategy: pick a known yardstick — usually a geometric series or p-series.

⚠️ Trap: direction matters. To prove convergence you must dominate by a larger convergent series. To prove divergence you must be dominated below by a divergent series.

3.5 Limit Comparison Test (LCT)

Hypotheses: $a_n, b_n > 0$.

Test: compute $\displaystyle L = \lim_{n\to\infty} \frac{a_n}{b_n}$.

Conclusion:

• $0 < L < \infty$ → $\sum a_n$ and $\sum b_n$ agree (both converge or both diverge)
• $L = 0$ and $\sum b_n$ converges → $\sum a_n$ converges
• $L = \infty$ and $\sum b_n$ diverges → $\sum a_n$ diverges

Why LCT often beats DCT: DCT requires a strict inequality you can prove; LCT only needs the same asymptotic order.

🎵 Audio 4 — Limit comparison — audio/ch3_04_limit_compare.mp3

Example

Decide $\displaystyle\sum_{n=1}^{\infty} \frac{2n+1}{n^3 + 5}$.

Step 1 — Pick $b_n$: leading-term analysis says $a_n \sim \frac{n}{n^3} = \frac{1}{n^2}$. Let $b_n = \frac{1}{n^2}$.

Step 2 — Compute the limit:
$$\lim_{n\to\infty} \frac{(2n+1)/(n^3+5)}{1/n^2} = \lim \frac{n^2(2n+1)}{n^3+5} = 2$$

$0 < 2 < \infty$. ✓

Step 3: $\sum \frac{1}{n^2}$ converges → original converges.

🔑 How to choose $b_n$: keep only the dominant term. For rational expressions, match the highest-power ratio.

3.6 Ratio Test (a must when factorials or exponentials appear)

When to use: terms contain $n!$, $r^n$, $n^n$, etc.

Compute:
$$L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$$

Conclusion:

• $L < 1$ → absolutely convergent
• $L > 1$ (including $L = \infty$) → divergent
• $L = 1$ → inconclusive, switch to a different test

🎵 Audio 5 — Ratio test + factorial cancellation — audio/ch3_05_ratio.mp3

Example 1 (factorial classic)

Decide $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}$.

$$L = \lim \left|\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}\right| = \lim \frac{(n+1) \cdot n^n}{(n+1)^{n+1}} = \lim \frac{n^n}{(n+1)^n} = \lim \frac{1}{(1+1/n)^n} = \frac{1}{e}$$

$L = \frac{1}{e} < 1$ → converges.

Example 2 (when $L=1$ fails)

Decide $\sum \frac{1}{n}$ (we already know it diverges).

$$L = \lim \frac{1/(n+1)}{1/n} = \lim \frac{n}{n+1} = 1$$

The test is inconclusive — use the integral test or the p-series rule instead.

🔑 Factorial cancellation: $\frac{(n+1)!}{n!} = n+1$ and $\frac{n!}{(n-1)!} = n$.

⚠️ Trap: ratio test always gives $L=1$ on a p-series. Don't waste time — use the p-series rule directly.

3.7 Root Test

When to use: the entire general term is raised to the $n$-th power, e.g. $\left(\frac{n}{n+1}\right)^n$.

Compute:
$$L = \lim_{n\to\infty} \sqrt[n]{|a_n|}$$

Conclusion: identical to ratio test ($<1$ converges, $>1$ diverges, $=1$ inconclusive).

🎵 Audio 6 — Root test — audio/ch3_06_root.mp3

Example

Decide $\displaystyle\sum_{n=1}^{\infty} \left(\frac{n}{2n+1}\right)^n$.

$$L = \lim \sqrt[n]{\left(\frac{n}{2n+1}\right)^n} = \lim \frac{n}{2n+1} = \frac{1}{2}$$

$L = \frac{1}{2} < 1$ → converges.

🔑 Ratio vs. root: an outer $n$-th power cries out for the root test; factorials cry out for the ratio test.

3.8 Quick-Reference Table

3.9 Mixed Examples

Decide convergence:

(a) $\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n}$

(b) $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 + \cos n}$

(c) $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$

(d) $\displaystyle\sum_{n=1}^{\infty} \frac{n+1}{n^3+2n}$

Solutions:

(a) Ratio test (because $2^n$):
$$L = \lim \frac{(n+1)^2/2^{n+1}}{n^2/2^n} = \frac{1}{2} \lim \left(\frac{n+1}{n}\right)^2 = \frac{1}{2} < 1$$
Converges.

(b) Direct comparison or limit comparison: for large $n$, $n^2 + \cos n \ge n^2 - 1$, so $\frac{1}{n^2 + \cos n} \le \frac{1}{n^2 - 1} \sim \frac{1}{n^2}$. Or: limit-compare with $\frac{1}{n^2}$, ratio $\to 1$. Converges.

(c) Integral test (because $\ln n$ in the denominator):
$$\int_2^{\infty} \frac{1}{x (\ln x)^2} dx \xrightarrow{u = \ln x} \int_{\ln 2}^{\infty} \frac{du}{u^2} = \frac{1}{\ln 2}$$
Integral finite → converges.

(d) Limit comparison with $\frac{1}{n^2}$: leading term is $\frac{n}{n^3} = \frac{1}{n^2}$. Converges.

🎵 Audio 7 — Walkthrough of (a)–(d) — audio/ch3_07_examples.mp3

3.10 Trap Alerts ⚠️

1. Don't push past $L=1$ in ratio/root — switch tests immediately.
2. Comparison/LCT only work for positive-term series. Take absolute values first if signs are mixed.
3. Integral test gives convergence, not the sum.
4. DCT direction matters: dominate above for convergence, dominate below for divergence.
5. LCT requires $0 < L < \infty$ for the two-way conclusion. $L = 0$ or $\infty$ gives a weaker one-way conclusion.
6. Memorize p-series and geometric series: nearly every comparison rests on them.

🎵 Audio 8 — Trap roundup — audio/ch3_08_traps.mp3

3.11 30-Second Decision Algorithm

1. Does a_n → 0? No → diverges (done)
2. Geometric or p-series? → apply formula
3. Has (-1)^n? → jump to Ch 4
4. Has n! or r^n? → Ratio test
5. Outer (...)^n? → Root test
6. Looks like $\frac{1}{n}$^p or r^n? → Comparison or LCT
7. f(n) positive, decreasing, easily integrable? → Integral test
8. None of the above? → try telescoping

🎵 Audio 9 — 30-second algorithm — audio/ch3_09_algorithm.mp3

3.12 Standard Yardsticks (memorize)

3.13 Mnemonic

"Recognize the shape, walk the decision tree."

• factorial / exponential → ratio
• outer $n$-th power → root
• easy-to-integrate function → integral
• looks like p-series or geometric → comparison / LCT
• $L = 1$ → switch tests immediately
• comparison only on positive terms

🎵 Audio 10 — Recap + mnemonic — audio/ch3_10_recap.mp3

Media Inventory

End of chapter. Next: alternating series and absolute/conditional convergence.