Series

A sequence lists its terms, whereas a series treats its terms as successive increments to be accumulated.

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

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

is called a series. Alternatively, we often write

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

for the series. In this situation, 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 $$s$,$ we call $$s$$ the sum of the series. Also we say that the series converges to $$s$$ and write

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

if $$(s_n)\to s$.$ Otherwise, if $$(s_n)$$ diverges, the series is said to diverge.

Indeed, every theorem about sequences can be stated in therms of series, and vice versa. In $$\mathbb C$,$ for example, the Cauchy criterion for convergence can be restated in the following form.

Theorem 4.1.2. A series $\sum_{n=1}^{\infty} x_n$ converges if and only if for any $$\varepsilon >0$,$ there exists $n_0\in \mathbb N^{+}$ 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.

Corollary 4.1.3. If a series $\sum_{n=1}^{\infty} x_n$ converges, then $$x_n\to 0$.$

References

Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.

External

4.1 Series

References