divergence to infinity
Some sequences fail to converge not by wobbling but by marching off the top of the chart — they grow past any number you care to name and never come back. This special, well-organized kind of divergence is called divergence to infinity, and although the sequence has no finite limit, its behavior is so regular that we write a_n -> +infinity.
Precisely, a_n -> +infinity means that for every real number M there exists an index N such that a_n > M for all n > N. In words: no matter how high a bar M you set, the sequence eventually clears it and stays above it. Symmetrically, a_n -> -infinity if for every M there is an N with a_n < M for all n > N. This mimics the epsilon-N definition, with 'beyond M' replacing 'within epsilon'.
A subtle but important point: a sequence that diverges to +infinity is divergent, not convergent — the symbol +infinity is shorthand for a pattern of growth, not a finite limit, and writing a_n -> +infinity does not mean the sequence converges. In the extended real number system one may treat +infinity as a genuine limit point, but in standard real analysis 'converges' is reserved for finite limits. Diverging to infinity is therefore a structured special case of divergence, sharply distinct from chaotic oscillation.
a_n = sqrt(n) -> +infinity: given M, take N > M^2; then for n > N we have sqrt(n) > sqrt(N) > M. The growth is slow, but it eventually exceeds and stays above every bound.
An M-N proof for infinite limits mirrors the epsilon-N proof for finite ones.
Unbounded does not imply divergence to infinity: the sequence 1, 0, 2, 0, 3, 0, ... is unbounded yet does not tend to +infinity, because it keeps dropping back to 0 and so never stays above any M.