What Is an Infinite Series?
Central Idea: An infinite series is not an act of adding infinitely many numbers one after another. It is the limit of a sequence of partial sums. The sum is a limit — and that single reframing dissolves every paradox about "adding infinitely many things."
Formal Definition
Given a sequence $\{a_n\}_{n=1}^{\infty}$, define the $N$-th partial sum
$$ S_N = a_1 + a_2 + \cdots + a_N = \sum_{n=1}^{N} a_n. $$The sequence $\{S_1, S_2, S_3, \ldots\}$ is an ordinary sequence — the mathematical citizen we already know how to study. We say the series $\displaystyle\sum_{n=1}^{\infty} a_n$ converges if and only if the sequence $\{S_N\}$ converges to a finite limit $S$:
$$ \sum_{n=1}^{\infty} a_n = S \;\;\Longleftrightarrow\;\; \lim_{N\to\infty} S_N = S. $$If $\{S_N\}$ diverges (goes to $\pm\infty$ or oscillates), we say the series diverges.
Callback to Sequences: Everything you learned about limits of sequences — squeeze theorem, monotone convergence, algebra of limits — now applies directly, because convergence of a series is convergence of the sequence $\{S_N\}$.
First Example: The Geometric Series
Consider $\displaystyle\sum_{n=0}^{\infty} ar^n$ with $a \neq 0$. The partial sum has a closed form (recall the geometric-progression formula):
$$ S_N = a\,\frac{1 - r^{N+1}}{1 - r}, \qquad r \neq 1. $$Now take the limit:
- If $|r| < 1$, then $r^{N+1} \to 0$, so $S_N \to \dfrac{a}{1-r}$. The series converges.
- If $|r| \ge 1$, then $r^{N+1}$ does not tend to $0$ (it grows or oscillates), so $\{S_N\}$ diverges. The series diverges.