Series — Introduction to Analysis

A sequence records a succession of states, whereas a series records successive increments and accumulates them. Some quantities in analysis arise in this way: a total effect is obtained by adding a sequence of smaller contributions. A series provides the language for studying whether this process approaches a finite value.

Definition 4.1.1. For a sequence $$(x_n)$$ in $$\mathbb C$,$ an expression of the form

\[x_1 + x_2 + x_3 + \cdots\]

is called a series. Alternatively, we often write $\sum_{n=1}^{\infty} x_n$ or $\sum x_n$ for the series. The number $$s_n$$ defined by

\[s_n = \sum_{k=1}^n x_k\]

is called the $$n$$ th partial sum of the series. If the sequence $$(s_n)$$ converges to $$s$,$ the series is said to converge to $$s$$ and we write

\[\sum_{n=1}^{\infty} x_n = s.\]

In this case, we call $$s$$ the sum of the series. Otherwise, if $$(s_n)$$ diverges, the series is said to diverge.

The passage between these two viewpoints is reversible. If we set $$s_0=0$,$ then

\[s_n=\sum_{k=1}^n x_k\quad \text{and} \quad x_n=s_n-s_{n-1}.\]

Thus the sequence of terms $$x_n$$ and the sequence of partial sums $$(s_n)$$ contain exactly the same information. Summation passes from successive increments to the accumulated state, while taking successive differences recovers the increments. In this precise sense, the theory of series is the theory of sequences expressed in terms of increments. The two viewpoints nevertheless emphasize different structures. Convergence and the Cauchy property concern the accumulated states $$s_n$,$ whereas positivity, size, and rearrangement are naturally expressed in terms of the increments $$x_n$.$ We first translate the Cauchy criterion for sequences into the language of increments.

Theorem 4.1.2. A series $\sum x_n$ converges if and only if for any $$\varepsilon >0$,$ there exists $n_0\in \mathbb Z^{+}$ such that $m\ge n \ge n_0$ implies

\[\left| \sum_{k=n}^{m} x_k \right| < \varepsilon.\tag{$\ast$}\]

The identity $\sum_{k=n}^{m}x_k=s_m-s_{n-1}$ makes the relation between this theorem and the Cauchy criterion transparent. A finite tail of the series measures the displacement between two partial sums, so $(\ast)$ says precisely that sufficiently late partial sums are arbitrarily close to one another. Taking $$m=n$$ also gives the following necessary condition.

Corollary 4.1.3. If a series $\sum x_n$ converges, then $$x_n$$ converges to $$0$.$

Remark 4.1.4. The condition $$x_n\to 0$,$ however, is not sufficient for the series to converge. For example, the series

\[\sum_{n=1}^{\infty}\frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots\]

diverges.

Positivity also has a simple interpretation in terms of series. Since $$x_n=s_n-s_{n-1}$,$ the condition $x_n\ge0$ means precisely that $$(s_n)$$ is increasing. Theorem 3.1.5, the monotone convergence theorem for real sequences, therefore yields the following result.

Theorem 4.1.5. A series of nonnegative terms in $\mathbb R$ converges if and only if its partial sums form a bounded sequence.

We now turn to a convergence test of a different nature, called the comparison test.

Theorem 4.1.6.

If $|x_n|\le y_n$ eventually and if $\sum y_n$ converges, then $\sum x_n$ converges.
If $x_n\ge y_n \ge 0$ eventually and if $\sum y_n$ diverges, then $\sum x_n$ diverges.