📂Analysis

Series, Infinite Series

Definition¹

Let’s assume a sequence $\left\{ a_{n} \right\}$ is given. Then, let’s define the following notation.

$$\sum \limits_{n=p}^{q} a_{n} = a_{p} + a_{p+1} + \cdots + a_{q}\quad (p \le q)$$

Define the partial sum $s_{n}$ of $\left\{ a_{n} \right\}$ as follows.

$$s_{n} = \sum \limits_{k=1}^{n} a_{k}$$

Then, we can think of a sequence $\left\{ s_{n} \right\}$ of these $s_{n}$. The limit of sequence $\left\{ s_{n} \right\}$ is called the infinite series or simply, the series of $\left\{ a_{n} \right\}$, and is denoted as follows.

$$\sum \limits_{n = 1}^{\infty} a_{n} = \lim \limits_{n \to \infty} s_{n} = \lim\limits_{n \to \infty}\sum \limits_{k=1}^{n} a_{k}$$

If $\left\{ s_{n} \right\}$ converges to $s$, it is denoted as follows and the series is said to converge.

$$\sum \limits_{n = 1}^{\infty} a_{n} = s$$

If $\left\{ s_{n} \right\}$ does not converge, the series is said to diverge. In the case of divergence,

• If for every $M \in \mathbb{R}$, there exists $N \in \mathbb{N}$ such that $n \ge N \implies s_{n} > M$ is satisfied, $$\sum \limits_{n = 1}^{\infty} a_{n} = \infty$$ it is denoted as follows.

• If for every $M \in \mathbb{R}$, there exists $N \in \mathbb{N}$ such that $n \ge N \implies x_{n} < M$ is satisfied, $$\sum \limits_{n = 1}^{\infty} a_{n} = -\infty$$ it is denoted as follows.

Explanation

A series is a rigorous mathematical definition of the ambiguous concept of adding up infinitely many terms. It is also denoted simply as $\sum a_{n}$.