Converge
A sequence or series converges when its terms or partial sums approach a specific finite value as the number of terms increases without bound.
Definition
A series converges when, as you keep adding terms, the running total gets closer and closer to a specific number without straying from it: formally, $\sum a_n$ converges if its sequence of partial sums $\{S_n\}$ has a finite limit $L$. For a geometric series $\sum a r^n$, convergence requires $|r| < 1$, with sum $a/(1-r)$, and while $\lim a_n = 0$ is necessary for convergence, it is not sufficient. In a metric space $(X, d)$, a series converges if its partial sums form a Cauchy sequence, equivalently if $\lim_{n \to \infty} S_n$ exists in $X$; absolute convergence ($\sum |a_n| < \infty$) implies convergence in complete (Banach) spaces, while conditional convergence (convergent but not absolute) permits rearrangement to any sum, the Riemann rearrangement theorem.
Example
$1 + 1/2 + 1/4 + 1/8 + \ldots$ converges to $2$: after $10$ terms the sum is already about $1.998$, approaching $2$ without ever exceeding it. $\sum (1/2)^n$ from $n=0$ converges since $|r| = 1/2 < 1$, summing to $1/(1 - 1/2) = 2$, and $\sum 1/n^2$ converges by the p-series test ($p = 2 > 1$) to $\pi^2/6$. The alternating harmonic series $\sum (-1)^{n+1}/n = 1 - 1/2 + 1/3 - \ldots$ converges conditionally to $\ln(2)$ by the alternating series test, yet rearranging its terms can yield any real value.
Key Insight
Convergence means "zeroing in on a target": the partial sums get closer and closer to the limiting value, like an arrow approaching a bulls-eye without ever quite reaching it, and several tests exploit different properties of the terms to check this: geometric ($|r| < 1$), p-series ($p > 1$), comparison, ratio, root, and integral. The Riemann rearrangement theorem shows that conditional convergence is "fragile," rearranging terms destroys the sum, while absolute convergence is the robust version, preserved under all rearrangements.