Chapter 21 — Key Takeaways
A compact recap of the chapter. If you can reconstruct each box below from memory, you are ready for the convergence tests of Chapter 22.
The Central Idea
A series is a limit, not an act of addition. An infinite series is defined as the limit of its sequence of partial sums:
$$\sum_{n=1}^\infty a_n = \lim_{N \to \infty} S_N, \qquad S_N = a_1 + a_2 + \cdots + a_N.$$
The series converges when this limit exists as a finite number, and diverges otherwise (whether the partial sums run to $\pm\infty$ or oscillate). Convergence of the series is convergence of the partial-sum sequence — every question about infinite addition routes back to the sequence machinery of Chapter 20. To recover a single term from the partial sums, use $a_n = S_n - S_{n-1}$. (§21.2)
The Key Series
| Series | Form | Sum | Converges when |
|---|---|---|---|
| Geometric (start $n=0$) | $\sum_{n=0}^\infty a r^n$ | $\dfrac{a}{1-r}$ | $\lvert r\rvert < 1$ |
| Telescoping | $\sum (b_n - b_{n+1})$ | $b_1 - \lim_{N\to\infty} b_{N+1}$ | $\lim b_n$ exists |
| Harmonic | $\sum_{n=1}^\infty \dfrac1n$ | $\infty$ | never (diverges) |
| p-series | $\sum_{n=1}^\infty \dfrac{1}{n^p}$ | $\zeta(p)$ when convergent | $p > 1$ |
- Geometric series is the most important series in the chapter: first term over one-minus-ratio, valid only for $|r| < 1$; it diverges for $|r| \ge 1$. Read $a$ off as the literal first term of the series as written. (§21.3)
- Telescoping series collapse because consecutive terms cancel; the usual tool for revealing the difference structure is partial fractions (Chapter 16). Watch the offset — a $b_n - b_{n+2}$ pattern leaves four survivors, not two. (§21.4)
- Harmonic series diverges even though $\tfrac1n \to 0$ — proved by Oresme's doubling-block grouping (each block exceeds $\tfrac12$). It grows like $\ln N$, so slowly that no numerical experiment could ever reveal the divergence. (§21.5)
- p-series sit on either side of the boundary $p = 1$; the harmonic series is the borderline case that just fails. (§21.8)
The Divergence Test — and Its Caveat
- Divergence test ($n$-th term test): if $\displaystyle\lim_{n\to\infty} a_n \neq 0$ (or the limit fails to exist), then $\sum a_n$ diverges. It is the cheapest check available — just look at the terms. (§21.6)
- The caveat (the most important caution in the chapter): $a_n \to 0$ does NOT imply convergence. The harmonic series is the standing counterexample. The test is one-way — it can only ever prove divergence. Never write "$a_n \to 0$, therefore the series converges." (§21.6)
Properties of Convergent Series
- Linearity: convergent series add and subtract term-by-term, and may be scaled by a constant: $\sum(a_n \pm b_n) = A \pm B$ and $\sum c\,a_n = cA$. (§21.7)
- The tail decides: changing finitely many terms changes the sum but never whether a series converges.
- What does NOT carry over: you cannot multiply two series term-by-term (the correct product is the Cauchy product), and you cannot freely rearrange terms unless the series converges absolutely — a distinction deferred to Chapter 22. (§21.7)
Memorable Facts and Applications
- Repeating decimals are geometric series: $0.\overline{3} = \sum \tfrac{3}{10^n} = \tfrac13$, and $0.\overline{9} = 1$ exactly (not merely "close to 1"). (§21.3)
- Present value of a perpetuity paying $C$ per year at rate $r$ is $C/r$ — a geometric series in disguise that underlies bond and stock pricing. (§21.3, §21.10)
- Drug accumulation / bouncing ball / multiplier effect: any process repeating with geometric decay (fraction $r$ retained per step) reaches the finite total $\dfrac{D}{1-r}$. (§21.10)
- Zeno's paradox dissolved: $\tfrac12 + \tfrac14 + \cdots = 1$, so infinitely many shrinking steps sum to a finite total. (§21.1, §21.10)
- Basel sum: $\sum 1/n^2 = \pi^2/6$ — convergent (it is a $p = 2$ series), derived in Chapter 24. (§21.8)
Common Errors to Avoid
- Writing "$a_n \to 0$, therefore $\sum a_n$ converges." (False — harmonic series.)
- Using $\sum r^n = \tfrac{1}{1-r}$ when the series starts at $n = 1$ (then the first term is $r$, and the sum is $\tfrac{r}{1-r}$). Always identify the actual first term $a$.
- Applying the geometric formula when $|r| \ge 1$ (the series diverges; there is no sum).
- Assuming "everything in the middle cancels" in a telescoping sum without writing out the first few and last few terms to check the offset.
- Concluding convergence/divergence of a $p$-series by guessing instead of comparing $p$ to $1$.
Connections and the Road Ahead
- Backward: sequences and their limit laws (Chapter 20); partial fractions for telescoping (Chapter 16); $a_n = S_n - S_{n-1}$ as the discrete echo of differentiating an accumulation (Chapter 14).
- Forward: Chapter 22 builds the full convergence toolkit (comparison, limit comparison, ratio, root, integral, and alternating series tests) and the absolute-vs-conditional distinction. Chapter 23 lets terms depend on $x$, giving power and Taylor series. Chapter 24 cashes those in for the Basel sum and Euler's identity.
A Reflection
Two facts organize everything to come, and both deserve memorization. First, a geometric series with $|r| < 1$ converges to $a/(1-r)$ — the seed of present value, repeating decimals, bouncing balls, drug plateaus, and Taylor series. Second, shrinking terms are necessary but never sufficient for convergence, with the harmonic series standing as the permanent counterexample. Keep these close: the rest of Part IV is the art of deciding any series, and it all stands on the foundation you built here.