Converge
Functions & Advanced AlgebraA 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 you keep adding terms and the running total gets closer and closer to a specific number without going past it.
Example
$1 + 1/2 + 1/4 + 1/8 + \ldots$ converges to $2$. After $1$ term: $1$. After $2$: $1.5$. After $3$: $1.75$. After $10$: about $1.998$. It approaches $2$ and never exceeds it.
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.
Definition
A series $\sum a_n$ converges if its sequence of partial sums $\{S_n\}$ has a finite limit $L$. For a geometric series $\sum a r^n$, convergence requires $|r| < 1$, and the sum equals $a/(1-r)$. The necessary condition for convergence is $\lim a_n = 0$, but this is not sufficient.
Example
$\sum (1/2)^n$ from $n=0$: $r = 1/2$, $|r| < 1$, so converges. Sum $= 1/(1 - 1/2) = 2$. The series $\sum 1/n^2$ converges by the p-series test ($p = 2 > 1$). Sum $= \pi^2/6$.
Key Insight
Several convergence tests exist: geometric ($|r| < 1$), p-series ($p > 1$), comparison, ratio, root, integral. Each exploits different properties of the terms to determine whether the total is finite.
Definition
A series $\sum a_n$ in a metric space $(X, d)$ converges if the sequence of partial sums is Cauchy, or equivalently if $\lim_{n \to \infty} S_n$ exists in $X$. Absolute convergence ($\sum |a_n| < \infty$) implies convergence in complete spaces (Banach spaces). Conditional convergence (converges but not absolutely) permits rearrangement to any sum (Riemann rearrangement theorem).
Example
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, but rearranging its terms can yield any real value.
Key Insight
The Riemann rearrangement theorem shows that conditional convergence is "fragile": rearranging terms destroys the sum. Absolute convergence is the robust version, preserved under all rearrangements.