What is $1 - 1 + 1 - 1 + \cdots$? Grouped one way, $(1 - 1) + (1 - 1) + \cdots = 0$. Grouped another, $1 - (1 - 1) - (1 - 1) - \cdots = 1$. Euler and Leibniz thought the “right” answer was $\frac{1}{2}$, the average. In the eighteenth century such sums were handled freely, and the habit sometimes led to absurd results.

Partial sums

In chapter 6 of the Cours d’analyse, Cauchy cut the problem off at the root. A series $u_0 + u_1 + u_2 + \cdots$ has partial sums

$$s_n = u_0 + u_1 + \cdots + u_{n-1}.$$

If $s_n$ approaches a limit $s$ as $n$ grows, the series is convergent and $s$ is its sum. If not, the series is divergent and, in Cauchy’s words, “has no sum”. For $1 - 1 + 1 - \cdots$ the partial sums jump between 1 and 0 forever: there is no sum, neither $\frac{1}{2}$ nor anything else.

Cauchy’s criterion

The trouble is that to use the definition you need to know what the limit is. Cauchy gave a test that looks only at the terms:

For the series to be convergent, it is necessary and sufficient that, for increasing values of n, the sums s(n+1) − s(n), s(n+2) − s(n), s(n+3) − s(n), … end up constantly taking numerical values smaller than any assignable limit.

— Augustin-Louis Cauchy, Cours d'analyse (1821)

That is: from some point on, all the partial sums lie within $\varepsilon$ of one another, however small $\varepsilon$ is. A sequence with this property is now called a Cauchy sequence.

With this criterion one sees at once why the harmonic series diverges, even though its terms tend to zero. From $s_n$ to $s_{2n}$ we add $n$ terms, all greater than or equal to $\frac{1}{2n}$:

$$s_{2n} - s_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \ge n \cdot \frac{1}{2n} = \frac{1}{2}.$$

However far you go, there are always two partial sums that differ by at least $\frac{1}{2}$. In the figure, choose a series and an $\varepsilon$, and see from where on the partial sums fit in a band of width $\varepsilon$:

Cauchy regarded the sufficiency of the criterion as obvious. To really prove it one had to know exactly what a real number is, and that was only done in 1872, by Dedekind and by Cantor. Curiously, Cantor built the real numbers precisely out of Cauchy sequences.

The root test

Cauchy also gave practical tests. The strongest is the root test: compute $\sqrt[n]{|u_n|}$ for large $n$. If these numbers end up below some $r < 1$, the series converges, because the terms become smaller than those of a geometric progression $r^n$. If they stay above 1, the terms do not go to zero and the series diverges. To deal with oscillating sequences, Cauchy introduced “the greatest of the limits”, now called the limit superior.

For example, for $\sum \frac{n}{2^n}$:

$$\sqrt[n]{\frac{n}{2^n}} = \frac{\sqrt[n]{n}}{2} \longrightarrow \frac{1}{2} < 1,$$

because $\sqrt[n]{n} \to 1$. The series converges (its sum is 2).

Exercise: the sum of n/2ⁿ

Compute the first four partial sums of $\sum_{n \ge 1} \frac{n}{2^n}$ and show that the sum is 2.

$s_1 = 0.5$, $s_2 = 0.5 + 0.5 = 1$, $s_3 = 1 + 0.375 = 1.375$, $s_4 = 1.375 + 0.25 = 1.625$. For the sum, call $S = \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \cdots$. Then $\frac{S}{2} = \frac{1}{4} + \frac{2}{8} + \cdots$, and subtracting,

$$S - \frac{S}{2} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1.$$

So $S = 2$. The calculation is legitimate because the root test has already guaranteed that the series converges.

Exercise: the condensation test

Another of Cauchy’s tests says that if the terms $u_n$ are positive and decreasing, $\sum u_n$ converges if and only if $\sum 2^k u_{2^k}$ converges. Use it to decide when $\sum \frac{1}{n^p}$ converges.

With $u_n = \frac{1}{n^p}$ we have $2^k u_{2^k} = \frac{2^k}{2^{kp}} = \left(2^{1-p}\right)^k$, a geometric progression with ratio $2^{1-p}$. It converges if and only if $2^{1-p} < 1$, that is, $p > 1$. So $\sum \frac{1}{n^2}$ converges and $\sum \frac{1}{n}$ diverges, as we had already seen.

Sources: Augustin-Louis Cauchy, Cours d’analyse (1821), chs. 6 and 9 · Niels Henrik Abel, “Untersuchungen über die Reihe 1 + m/1 x + …”, Journal für die reine und angewandte Mathematik 1 (1826) · Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus (1981) · Umberto Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass (1986).

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