Chapter 20 — Key Takeaways

A compact recap of the discrete side of calculus: what a sequence is, when it converges, and the tools that decide. Use this as a pre-exam checklist and as a bridge into series (Chapter 21).

1. A Sequence Is a Function on $\mathbb{N}$

  • A sequence is a function $a:\mathbb{N}\to\mathbb{R}$, written $\{a_n\}$, where $a_n$ is the value at index $n$. It is just a function sampled at the integers — the discrete analog of $f(x)$. (§20.1)
  • Sequences arrive in three disguises: explicit ($a_n$ as a formula in $n$), recursive ($a_{n+1} = f(a_n)$ from a starting value), and implicit enumeration ("the $n$-th such object"). Explicit and recursive forms dominate calculus. (§20.2)

2. Convergence and Divergence

  • $\{a_n\}$ converges to $L$, written $a_n \to L$, if for every $\varepsilon > 0$ there is an integer $N$ with $n \ge N \implies |a_n - L| < \varepsilon$. This is the ε-N definition — Chapter 3's limit-at-infinity with the continuous threshold replaced by a cutoff index $N$. (§20.3)
  • A sequence is divergent if no such finite $L$ exists: it may run to $\pm\infty$, oscillate (like $(-1)^n$), or wander. Bounded is not the same as convergent — the terms must approach a single value. (§20.3)

3. Limit Laws and the Squeeze Theorem

  • Sequence limits inherit the limit laws of Chapter 3: sums, differences, scalar multiples, products, quotients (denominator limit $\neq 0$), and absolute values all pass through the limit. (§20.4)
  • The Continuous-Function Rule: if $g$ is continuous at $L$ and $a_n \to L$, then $g(a_n) \to g(L)$ — you may pass a limit inside a continuous function. (§20.4)
  • The Squeeze Theorem: if $b_n \le a_n \le c_n$ eventually and $b_n \to L$, $c_n \to L$, then $a_n \to L$. This is the clean tool for taming oscillations, e.g. $\dfrac{\sin n}{n}\to 0$. (§20.4)

4. Standard Limits and the Growth Hierarchy

  • Memorize the catalog of §20.5: $\dfrac{1}{n^p}\to 0$ ($p>0$); $r^n \to 0$ for $|r|<1$; $n^{1/n}\to 1$; $a^{1/n}\to 1$ ($a>0$); $\dfrac{\ln n}{n^p}\to 0$; and $\left(1 + \dfrac{x}{n}\right)^n \to e^x$. (§20.5)
  • Behind the catalog is one ordering — the growth hierarchy: $\;\ln n \prec n^p \prec r^n \prec n! \prec n^n\;$ ($p>0$, $r>1$). In any ratio of these, the faster-growing one wins, deciding the limit. (§20.5)

5. The Monotone Convergence Theorem

  • A sequence is monotone if it is non-decreasing or non-increasing, and bounded if it stays within fixed numbers. (§20.6)
  • Monotone Convergence Theorem (MCT): a bounded monotone sequence converges (non-decreasing + bounded above $\to$ converges to its supremum; non-increasing + bounded below $\to$ converges to its infimum). (§20.6)
  • The MCT's superpower: it certifies convergence without revealing the limit. You check monotonicity and boundedness — two often-tractable facts — and get convergence for free. Its truth rests on the completeness of $\mathbb{R}$. (§20.6)

6. Recursive Sequences and Fixed Points

  • A recursion $a_{n+1} = f(a_n)$ with $f$ continuous can converge only to a fixed point $L = f(L)$. So finding the limit reduces to solving $f(x) = x$. (§20.7)
  • Stability criterion: a fixed point $a^*$ is attracting when $|f'(a^*)| < 1$ (errors shrink, $e_{n+1}\approx f'(a^*)e_n$) and repelling when $|f'(a^*)| > 1$. (§20.7)
  • Newton's method is the recursion $x_{n+1} = x_n - g(x_n)/g'(x_n)$; its fixed points are roots of $g$, and $f'(a^*)=0$ at a simple root gives quadratic convergence — correct digits roughly double each step. The Babylonian method for $\sqrt2$ is Newton on $g(x)=x^2-2$. (§20.6–20.7)

7. A Sequence Is Not a Series

  • A sequence is a list $a_1, a_2, a_3, \ldots$; a series is a sum $a_1 + a_2 + a_3 + \cdots$ (Chapter 21). They are different objects.
  • The terms going to $0$ does not make the sum finite: $a_n = 1/n \to 0$, yet the harmonic series $\sum 1/n$ diverges (Chapter 21). Confusing "terms $\to 0$" with "sum is finite" is the most common error in all of Part IV. (§20.9)

Common Errors to Avoid

  • Calling $(-1)^n$ convergent because it is bounded. Bounded $\ne$ convergent. (§20.3)
  • Writing "$L = f(L)$, therefore the sequence converges." The fixed-point equation gives a candidate limit only; convergence must be established separately via the MCT or $|f'(a^*)|<1$. (§20.7)
  • Treating the sequence $a_n$ and the series $\sum a_n$ as the same convergence question. (§20.9)
  • Forgetting that $\sin(\pi n) = 0$ for integers (it converges) even though $\sin(\pi x)$ has no limit at infinity — sequences can be better-behaved than their continuous parents. (§20.3)

Connections

  • Backward: the ε-N definition is the ε-δ limit of Chapter 3; the Continuous-Function Rule is the sequential view of continuity (Chapter 4); Newton's method and quadratic convergence come from Chapter 11.
  • Forward: a series (Chapter 21) is the limit of its partial-sum sequence; the convergence tests (Chapter 22) are theorems about sequences; Taylor series (Chapter 23) converge when a sequence of polynomial approximations does. Mastering sequence convergence is the master key to Part IV.

What's Next

Chapter 21 takes the step from lists to sums, defining a series as the limit of its sequence of partial sums and meeting the geometric series (convergent) and the harmonic series (divergent terms-to-zero cautionary tale). Chapter 22 builds the full battery of convergence tests. Chapter 23 reaches power and Taylor series. Every one of those rests on the single question you can now answer: does this sequence converge?