Diverge
A sequence or series diverges when its terms or partial sums do not approach a finite limit.
Definition
A series diverges when the running total does not approach a fixed number: it might grow forever, shrink forever, or oscillate without settling. Types of divergence include unbounded growth ($1 + 2 + 3 + \ldots$), oscillation ($1 - 1 + 1 - \ldots$), or bounded but non-convergent behavior, and for geometric series divergence occurs when $|r| \ge 1$. Formally, a series $\sum a_n$ diverges if its partial sums do not converge in the given topology; regular summability methods (Cesaro, Abel, Ramanujan) can nonetheless assign values to some divergent series in a consistent way, extending the concept of "sum," with Ramanujan summation famously assigning $1 + 2 + 3 + \ldots = -1/12$, meaningful in the context of zeta function regularization.
Example
$1 + 2 + 3 + 4 + \ldots$ diverges because the sum keeps growing forever, and $1 - 1 + 1 - 1 + \ldots$ also diverges because the partial sums oscillate between $0$ and $1$, never settling. The harmonic series $1 + 1/2 + 1/3 + 1/4 + \ldots$ diverges despite its terms approaching $0$: grouping as $1 + (1/2) + (1/3+1/4) + (1/5+\ldots+1/8) + \ldots$ shows each group exceeds $1/2$, so the sum is unbounded. In string theory and quantum field theory, zeta regularization assigns finite values to divergent series to extract physical predictions, for instance $\zeta(-1) = -1/12$ "sums" the natural numbers.
Key Insight
Divergence means "no landing spot": the running total never zeroes in on a specific number, so not every infinite addition problem has an answer. The divergence test says that if $\lim a_n \neq 0$ the series definitely diverges, but if $\lim a_n = 0$ it might converge or diverge, the harmonic series being the classic example of the latter. Regularization of divergent series is not mathematical trickery but a well-defined extension of summability, and understanding the precise conditions under which a regularization is consistent remains an active area of analysis.