Series

An infinite sum $a_0 + a_1 + a_2 + \cdots$ is not a sum at all in the everyday sense: nobody can add infinitely many numbers. It is a limit, the limit of the finite sums $a_0$, $a_0 + a_1$, $a_0 + a_1 + a_2$, …, and everything from 2A.6 Sequences applies to it. That simple reduction hides two surprises. The terms can shrink to zero while the sum still grows without bound. And for some series, adding the same terms in a different order gives a different answer, or any answer you like.

Series run through the rest of the guidebook. Fourier series solve the heat equation (2B.7 Fourier Series and the First Heat Equation). Power series define the exponential function (2B.6 Power Series, Exponentials and Bump Functions). The geometric series inverts operators of the form $I - A$ (4A.1 Banach Spaces and Bounded Operators), a key step in proving that equations like Ricci flow have solutions at all (11A.3 Short-Time Existence and Uniqueness).

By the end of this chapter you will be able to:

• decide whether a series converges, using the Cauchy criterion, comparison, condensation, and the root, ratio and alternating series tests;
• explain why $\sum 1/n$ diverges while $\sum 1/n^2$ converges;
• use the geometric series as a sum and as an inverse;
• explain absolute and conditional convergence, and what rearrangement does to each;
• apply these to real problems: book stacking, drug dosing, and why the order of summation matters on a computer.

7.1Finite sums

For real numbers $a_m, \dots, a_n$, the sum $\sum_{i=m}^n a_i$ is defined by recursion (2A.1 The Natural Numbers): it is $0$ if $n < m$, and $\sum_{i=m}^{n+1} a_i = \big(\sum_{i=m}^n a_i\big) + a_{n+1}$. The familiar properties are proved by induction on the number of terms: sums split, $\sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i = \sum_{i=m}^{p} a_i$; they are linear, $\sum (a_i + b_i) = \sum a_i + \sum b_i$ and $\sum c\,a_i = c\sum a_i$; they preserve inequalities; and the triangle inequality extends, $\big|\sum a_i\big| \leq \sum |a_i|$.

Two further facts about finite sums are less obvious than they look, and both fail for infinite sums in general.

• Order doesn't matter. For a finite set $X$ and a function $f : X \to \mathbb{R}$, the sum $\sum_{x \in X} f(x)$ can be computed by listing $X$ in any order, and every order gives the same answer. Tao proves this carefully by induction on the size of $X$.
• Double sums can be swapped. $\sum_{i=1}^n \sum_{j=1}^m a_{ij} = \sum_{j=1}^m \sum_{i=1}^n a_{ij}$: summing a finite table by rows or by columns gives the same total.

7.2Infinite series

Applying the Cauchy criterion (2A.6 Sequences) to the partial sums gives a test that never mentions the sum.

In words: blocks of terms far out in the series must add up to almost nothing. Taking $p = q$ gives a necessary condition.

The converse is false, and the standard example is the most important divergent series in mathematics.

This argument is due to Nicole Oresme, writing around 1350. It shows something quantitative: the partial sum $H_n = 1 + \tfrac12 + \cdots + \tfrac1n$ grows like $\log_2 n / 2$ at least. In fact $H_n$ grows like the natural logarithm, $H_n \approx \ln n + 0.5772$; the constant $0.5772\ldots$ is the Euler–Mascheroni constant (2A.11 The Riemann Integral). So the growth is extremely slow. You need about $12{,}367$ terms to pass $10$, and more than $10^{43}$ to pass $100$.

7.3The geometric series

The formula $1 + x + x^2 + \cdots = \frac{1}{1-x}$ has two lives. Read left to right, it evaluates a sum. Read right to left, it inverts something: it writes $(1 - x)^{-1}$ as a series. That second reading generalises far beyond numbers. In 4A.1 Banach Spaces and Bounded Operators the same series, with $x$ replaced by a linear operator $A$ of size less than $1$, gives the inverse $(I - A)^{-1} = I + A + A^2 + \cdots$ (the Neumann series). This is how one shows that equations close to a solvable one are themselves solvable, an idea at the heart of short-time existence for Ricci flow (11A.3 Short-Time Existence and Uniqueness).

7.4Series of non-negative terms

If every $a_n \geq 0$, the partial sums increase, so by monotone convergence (2A.6 Sequences) the series converges exactly when its partial sums are bounded. Otherwise it diverges to $+\infty$. This makes non-negative series much easier to handle, and gives two powerful tests.

So $\sum 1/n^2$ converges. Euler found its sum in 1734: $\frac{\pi^2}{6} \approx 1.6449$. The dividing line between convergence and divergence sits exactly at $p = 1$. The dimension-counting arguments of later courses produce exactly this kind of sharp threshold: whether an integral like $\int |x|^{-p}\,dx$ is finite near $0$ depends on $p$ compared with the dimension, which decides when Sobolev functions are continuous (4A.10 Sobolev Embeddings and Critical Exponents) and when a singularity is integrable.

7.5Absolute and conditional convergence

Conditionally convergent series exist. The simplest test produces them.

So $1 - \tfrac12 + \tfrac13 - \tfrac14 + \cdots$ converges. Its sum is $\ln 2 \approx 0.6931$ (2B.6 Power Series, Exponentials and Bump Functions). But it is only conditionally convergent: the absolute values form the harmonic series. The difference turns out to matter enormously.

7.6Rearrangements

A rearrangement of $\sum a_n$ is a series $\sum a_{f(n)}$, where $f : \mathbb{N} \to \mathbb{N}$ is a bijection: the same terms, each used exactly once, in a different order. For finite sums, order never matters. For infinite ones, it depends on the kind of convergence.

For conditionally convergent series the situation is the opposite, and startling.

The theorem is due to Bernhard Riemann, in his 1854 habilitation thesis on trigonometric series, which was published after his death. The practical moral is that a conditionally convergent sum is fragile: its value depends on the order in which its terms are added, and any manipulation that reorders terms must be justified by absolute convergence.

7.7The root and ratio tests

For series that behave roughly like geometric series, two tests compare them with one directly.

Neither test decides anything when the relevant limit equals $1$: both $\sum 1/n$ and $\sum 1/n^2$ have ratio tending to $1$, but one diverges and the other converges. The root test is the one that determines the radius of convergence of a power series $\sum c_n x^n$, the interval of $x$ for which it converges (2B.6 Power Series, Exponentials and Bump Functions).

7.8Exercises

Solution

The partial sums telescope: $\sum_{n=1}^N \big(\tfrac1n - \tfrac1{n+1}\big) = 1 - \tfrac{1}{N+1} \to 1$. For $n \geq 2$, $\tfrac1{n^2} \leq \tfrac{1}{(n-1)n}$, so $\sum_{n \geq 1} \tfrac1{n^2} \leq 1 + \sum_{n \geq 2}\tfrac{1}{(n-1)n} = 1 + 1 = 2$.

Solution

(a) By induction: if the top $k - 1$ books have their combined centre of mass at the right edge of book $k$, then adding book $k$ (centre of mass at its middle, $\tfrac12$ to the left of that edge) moves the combined centre of the top $k$ books to $\tfrac{(k-1)\cdot 0 + 1 \cdot(-\tfrac12)}{k} = -\tfrac1{2k}$ relative to that edge. That is how far book $k$ can overhang the book below. (b) $\tfrac12 H_n > 3$ needs $H_n > 6$, so $\ln n > 5.4228$, giving $n \approx 227$. A direct computation gives $H_{226} \approx 5.99996$ and $H_{227} \approx 6.00437$, so $227$ books.

Solution

(a) Converges absolutely (ratio $\tfrac12$). (b) Converges conditionally (alternating test; $\sum 1/\sqrt n$ diverges). (c) Diverges, by condensation: $\sum 2^k \cdot \tfrac{1}{2^k k \ln 2} = \tfrac{1}{\ln 2}\sum \tfrac1k$. (d) Converges absolutely: the ratio is $(1 + 1/n)^{-n} \to 1/e < 1$ (2A.6 Sequences). (e) Converges absolutely, since $0 \leq \sin x \leq x$ for $x \geq 0$, so compare with $\sum 1/n^2$.

Solution

$r = (\tfrac12)^{12/8} = 2^{-1.5} \approx 0.354$. Accumulation ratio $\tfrac{1}{1 - r} \approx 1.55$. After $n$ doses the peak is a fraction $1 - r^n$ of steady state; $r^n \leq 0.05$ needs $n \geq \ln 0.05 / \ln 0.354 \approx 2.88$, so $3$ doses.