Definition

An infinite series is what you get when you add up infinitely many numbers following a pattern; sometimes the total approaches a specific finite number, and sometimes it grows forever. Formally, $\sum_{n=1}^{\infty} a_n$ is defined as the limit of its partial sums: if $\lim_{n \to \infty} S_n = L$, the series converges to $L$, otherwise it diverges, and while the terms $a_n$ must approach $0$ for convergence, this alone is not sufficient. More generally, an infinite series is a formal expression $\sum a_n$ whose value, if it exists, is $\lim S_n$ in the underlying metric space; in a Banach space, absolute convergence ($\sum \|a_n\| < \infty$) implies convergence, and power series $\sum a_n x^n$ converge on open disks in $\mathbb{C}$, representing analytic functions within their radius of convergence.