Infinite Series

Infinite Series

Definition 22.1. Let $\{a_n\}$ be a sequence. For each positive integer $n$, let $$s_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^{n} a_k.$$ An infinite series is the ordered pair of sequences $(\{a_n\}, \{s_n\})$.
$s_n$ is called the $n$th partial sum of the infinite series. 

Convergent Series

Definition 22.2. Let $\sum_{n=1}^{\infty} a_n$ be an infinite series. If the sequence of partial sums $\{s_n\}$ ($s_n = a_1 + \cdots + a_n$) converges to $L$, we say that the infinite series $\sum_{n=1}^{\infty} a_n$ converges to $L$ or that the infinite series $\sum_{n=1}^{\infty} a_n$ has sum $L$. If the sequence $\{s_n\}$ diverges, we say that the infinite series $\sum_{n=1}^{\infty} a_n$ diverges.

Theorem 22.3

Theorem 22.3. If the infinite series $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n \to \infty} a_n = 0$.

Proof. Let $\{ s_n \}$ be a partial sum of the infinite series. Let $\sum_{n=1}^{\infty} a_n = L = \lim_{n \to \infty} s_n$. Since $s_n - s_{n-1} = a_n$ for all positive integers $n$ (we define $s_0 = 0$), $$\lim_{n \to \infty} (s_n - s_{n-1}) = \lim_{n \to \infty} a_n = L - L = 0. \blacksquare$$

The Geometric Series

Theorem 22.4 (The Geometric Series). Let $a$ be a nonzero number. Then (i) $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1 - r}$ if $|r| < 1$.
(ii) $\sum_{n=0}^{\infty} ar^n$ diverges if $|r| \geq 1$. ($r^0$ is defined to be 1.) 

Proof. (i) 

Theorem 23.1

Theorem 23.1. Let $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ be series, and let $c$ be a real number. If $\sum_{n=1}^{\infty} a_n$ converges to $L$ and $\sum_{n=1}^{\infty} b_n$ converges to $M$, then $\sum_{n=1}^{\infty} (a_n + b_n)$ converges to $L + M$, and $\sum_{n=1}^{\infty} ca_n$ converges to $cL$.

Proof. Let $\{ s_n \}$ and $\{ t_n \}$ be the partial sums of $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$. Then $\{ s_n + t_n \}$ is the partial sum of $\sum_{n=1}^{\infty} (a_n + b_n)$. By definition 22.2 and the assumption, $$\lim_{n \to \infty} s_n = L, \lim_{n \to \infty} t_n = M \\ \Longrightarrow \sum_{n=1}^{\infty} (a_n + b_n) = \lim_{n \to \infty} (s_n + t_n) = L + M.$$ And similarly, we have $$\sum_{n=1}^{\infty} ca_n = \lim_{n \to \infty} cs_n = cL. \blacksquare$$