Chapter 8 — Sequences and Series
Adding up infinitely many numbers — and, surprisingly often, getting a finite answer like $\tfrac12+\tfrac14+\tfrac18+\cdots=1$.
About 2400 years ago the Greek philosopher Zeno of Elea posed a puzzle. To walk to a wall one metre away you must first cover half the distance, then half of what is left, then half of what is left again, and so on, forever. That is infinitely many tasks: $\tfrac12+\tfrac14+\tfrac18+\tfrac1{16}+\cdots$ metres. How can anyone finish infinitely many tasks in a finite time? Zeno concluded that motion is an illusion. The way out is that an infinite sum can have a finite value: the distances get smaller fast enough that their total is exactly $1$ metre, and, at constant speed, the times add up to a finite total too.
Making sense of "infinitely many additions" is the subject of this chapter. We first study sequences — endless lists $a_1,a_2,a_3,\dots$ — and what it means for them to settle down to a limit. Then we add the terms of a sequence to form an infinite series $\sum a_n$, and develop a toolkit of convergence tests that decide whether the sum is finite without having to find it. This is not just a curiosity: calculators evaluate $e^x$, $\sin x$ and $\ln x$ by adding up series; engineers, physicists and economists use them to approximate functions, solve differential equations and price long-term payments. Chapter 9 (power series) is built entirely on the tools you will learn here.
- find the limit of a sequence (or show it diverges) using limit laws, L'Hôpital's rule, the squeeze theorem and the Monotone Convergence Theorem;
- explain what it means for an infinite series to converge, in terms of its partial sums;
- sum geometric and telescoping series exactly, and use them in applications such as repeated drug doses and repeating decimals;
- decide whether a series converges using the divergence, integral, comparison, limit comparison, alternating series, ratio and root tests — and choose the right test quickly;
- distinguish absolute from conditional convergence and estimate the error when a sum is cut off after finitely many terms.
A first look: which test when?
Most of this chapter answers one question: does $\sum a_n$ converge? You will meet seven convergence tests and a few special series we can settle outright, each suited to a particular look of $a_n$. Here is the map you will be filling in; do not worry about the details yet — each row is explained carefully in its own section.
| If the terms $a_n$ look like… | try… | what it tells you | section |
|---|---|---|---|
| anything — check first! | Divergence test | if $a_n\not\to0$, the series diverges | 8.3 |
| $ar^{n-1}$ (constant ratio) | Geometric series | converges to $\dfrac{a}{1-r}$ exactly when $|r|\lt1$ | 8.2 |
| $b_n-b_{n+1}$ (partial fractions) | Telescoping | $S_N$ collapses; take its limit | 8.2 |
| $\dfrac1{n^p}$ | $p$-series | converges exactly when $p\gt1$ | 8.3 |
| $f(n)$ with $f$ positive, decreasing, easy to integrate | Integral test | $\sum a_n$ and $\int_1^\infty f(x)\,dx$ behave alike | 8.3 |
| rational or algebraic in $n$, e.g. $\dfrac{n+1}{n^3+5}$ | (Limit) comparison | compare with a $p$-series or geometric series | 8.4 |
| $(-1)^nb_n$ with $b_n$ decreasing to $0$ | Alternating series test | converges, with an easy error bound | 8.5 |
| factorials $n!$, powers $c^n$, products | Ratio test | look at $\left|\dfrac{a_{n+1}}{a_n}\right|$ | 8.6 |
| an $n$th power, e.g. $\left(\dfrac{2n}{3n+1}\right)^n$ | Root test | look at $\sqrt[n]{|a_n|}$ | 8.6 |
Sections in this chapter
Sequences
Explicit and recursive sequences, the ε–N meaning of a limit, the squeeze theorem, standard limits and the Monotone Convergence Theorem.
Infinite Series
Partial sums, geometric and telescoping series, the divergent harmonic series and applications.
The Divergence and Integral Tests
Terms that do not shrink to zero; comparing a sum with an improper integral; $p$-series and remainder estimates.
Comparison Tests
Decide convergence by comparing with a series you already understand.
Alternating Series
Signs that alternate, the alternating series test, error bounds, absolute and conditional convergence.
Ratio and Root Tests
Tests for factorials and powers, and a strategy for choosing among all the tests.