Chapter 22 — Key Takeaways

This is the hardest single chapter in the book, and almost all of the difficulty is choosing the right test. So we lead with the decision framework, then state each test precisely, then collect the facts worth memorizing.

The Decision Framework (which test, when)

Run a brand-new series $\sum a_n$ down this list in order; the first matching rule usually wins. This is the condensed form of §22.10 — the single most valuable thing in the chapter.

The one habit to build: name the test before you compute. Experts are not faster at any single test — they are faster at recognizing which test the series is asking for.

Each Test, Stated Precisely

Divergence test (§22.1). If $a_n \not\to 0$, then $\sum a_n$ diverges. (If $a_n \to 0$, the test says nothing.)

Integral test (§22.2). If $f$ is positive, continuous, and decreasing on $[1,\infty)$ with $a_n=f(n)$, then $\sum a_n$ and $\int_1^\infty f(x)\,dx$ (an improper integral, Chapter 17) converge or diverge together. Yields the $p$-series rule.

Direct comparison test (§22.3). If $0\le a_n\le b_n$: $\sum b_n$ converges $\Rightarrow \sum a_n$ converges; $\sum a_n$ diverges $\Rightarrow \sum b_n$ diverges. (Bound above by convergent for convergence; below by divergent for divergence.)

Limit comparison test (§22.4). For $a_n,b_n>0$ with $\lim a_n/b_n = c$: if $0

Ratio test (§22.5). $\rho=\lim|a_{n+1}/a_n|$: $\rho<1$ converges absolutely; $\rho>1$ diverges; $\rho=1$ inconclusive. Built for factorials and exponentials.

Root test (§22.6). $\rho=\lim|a_n|^{1/n}$: same verdicts as the ratio test. Built for $n$-th powers; strictly stronger than the ratio test.

Alternating series test / Leibniz (§22.7). For $\sum(-1)^{n+1}b_n$ with $b_n>0$: if $b_n$ is eventually decreasing and $b_n\to 0$, the series converges.

The Alternating Series Error Bound

For a convergent alternating series satisfying both Leibniz conditions, $$\boxed{\;|L - S_N| \le b_{N+1}\;}$$ The truncation error is at most the first omitted term, and it carries that term's sign. To approximate $L$ to within $\varepsilon$, sum until $b_{N+1}<\varepsilon$. This certified, computable bound is what makes alternating tails prized in numerical computing (Case Study 1). It applies only when the signs alternate and $b_n$ is decreasing to $0$ — verify both first.

Absolute vs. Conditional Convergence (§22.8)

• Absolutely convergent: $\sum|a_n|$ converges. The stronger condition; lets you apply positive-term tests to mixed signs, and the sum is order-independent.
• Conditionally convergent: $\sum a_n$ converges but $\sum|a_n|$ diverges. Convergence is borrowed from sign cancellation; the sum is order-dependent.
• Absolute Convergence Theorem: $\sum|a_n|$ converges $\Rightarrow \sum a_n$ converges.
• Riemann Rearrangement Theorem: a conditionally convergent series can be rearranged to sum to any real number, or to diverge. Infinite addition is not commutative.

Classification recipe: if the series converges, test $\sum|a_n|$. Converges → absolute. Diverges → conditional.

Facts Worth Memorizing

• $p$-series: $\sum 1/n^p$ converges $\iff p>1$ (your universal limit-comparison yardstick).
• Geometric: $\sum r^n$ converges $\iff |r|<1$, with sum $\tfrac{1}{1-r}$ (the model behind the ratio and root tests).
• Harmonic $\sum 1/n$ diverges ($p=1$); the alternating harmonic $\sum(-1)^{n+1}/n=\ln 2$ converges (conditionally).
• $\sum 1/n^2 = \pi^2/6$ (Basel); $\sum_{n\ge0} 1/n! = e$, so $\sum_{n\ge1}1/n! = e-1$.
• Leibniz: $\sum_{n\ge1}(-1)^{n+1}/(2n-1) = \pi/4$.

Common Errors

• Treating $a_n\to 0$ as sufficient. It is only necessary. $\sum 1/n$ has $a_n\to 0$ yet diverges.
• Comparison in the wrong direction. To show convergence, bound above by a convergent series; to show divergence, bound below by a divergent one. Comparing $1/(n^2+1)$ to the larger, divergent $1/n$ proves nothing.
• Using the ratio/root test when $\rho=1$. Inconclusive — every $p$-series gives $\rho=1$. Switch to integral test or limit comparison.
• Applying the error bound to a non-alternating series, or before confirming $b_n$ decreases to $0$. Neither is valid.
• Forgetting the integral test needs $f$ decreasing (continuity and positivity too).
• Rearranging a conditionally convergent series as if order were irrelevant — it changes the sum (Riemann).

Connections

• Backward: the integral test rests on improper integrals (Chapter 17); geometric and telescoping series and the divergence test come from Chapter 21; bounded-monotone convergence of partial sums is Chapter 20.
• Forward: Chapter 23 applies the ratio and root tests to power series $\sum c_n x^n$ to find the radius of convergence, and the alternating-series error bound to certify how well a finite Taylor polynomial approximates a function — finishing the area-under-the-normal-curve thread from Chapter 13.
• Theme: Approximation is the soul of calculus — every convergent series is an infinite approximation, and the rate (set by $\rho$) decides whether it is computable in practice. Hand computation builds understanding; machine computation builds power — apply each test by hand, then watch Python confirm the rate (§22.9).

Reflection

The densest chapter in Part IV: seven tests with overlapping applicability, where the real skill is recognition, not execution. That skill develops only through volume — work many series, run each through the framework above, and by the time you reach Chapter 23 the convergence questions will feel mechanical, freeing you to focus on what the answer means.