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 $$\sum_{n=1}^{\infty} ca_n = \lim_{n \to \infty} cs_n = cL. \blacksquare$$