Definition 1. Given a sequence of numbers $\{ a_n \}$, an expression of the form $$a_1 + a_2 + \cdots + a_n + \cdots$$ is an infinite series. The number $a_n$ is the $n$th term of the series. The sequence $\{ s_n \}$ defined by $$s_n = \sum_{k=1}^n a_k$$ is the sequence of partial sums of the series, the number $s_n$ being the $n$th partial sum. If the sequence of partial sums converges to a limit $L$, we say that the series converges and that its sum is $L$. In this case, we also write $$a_1 + a_2 + \cdots + a_n + \cdots = \sum_{n=1}^{\infty} a_n = L.$$ If the sequence of partial sums of the series does not converge, we say that the series diverges.
Geometric Series
Definition 2.Geometric series are series of the form $$a + ar + ar^2 + \cdots + ar^{n-1} + \cdots = \sum_{n=1}^{\infty} ar^{n-1}$$ in which $a$ and $r$ are fixed real numbers and $a \neq 0$. The number $r$ is called the ratio.
Theorem 1. For a geometric series $a + ar + \cdots + ar^{n-1} + \cdots$, if $|r| < 1$, the series converges to $\frac{a}{1-r}$: $$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}, \text{ } |r| < 1.$$ If $|r| \geq 1$, the series diverges.
Properties
Theorem 2. If $\sum a_n = A$ and $\sum b_n = B$ are convergent series, then (1) $\sum (a_n \pm b_n) = \sum a_n \pm \sum b_n = A \pm B$ (2) $\sum ka_n = k \sum a_n = kA$ for any number $k$.
Corollary.(1) Every nonzero constant multiple of a divergent series diverges. (2) If $\sum a_n$ converges and $\sum b_n$ diverges, then $\sum (a_n \pm b_n)$ diverges.
Theorem
Theorem 2. A series $\sum_{n=1}^{\infty} a_n$ of nonnegative terms converges if and only if its partial sums are bounded above.