convergent,
adj. 1. (of an infinite sequence of numbers or vectors) having a finite limit, so that if an is the nth element of the sequence, there exists an L, the limit, such that for every ɛ > 0 there is an N such that
| an - L | < ɛ for all n > N.
A corresponding definition applies in a metric space setting. 2. (of an infinite series) having a finite sum, generating a sequence of partial sums that has a finite limit. If the series a0 + a1 + a2 + ..., that is, the sequence of partial sums
〈a0, a0 + a1, a0 + a1 + a2,...〉,
converges, then the sequence 〈a0, a1, a2,... 〉 must converge to zero, but not necessarily conversely. For example, the sequence
〈1, 1/2, 1/3, 1/4,...〉
is convergent, but the series
1 + 1/2 + 1/3 + 1/4 + ...
is not. 3. (of an improper integral) having a finite value that is defined as the limit of the proper integrals as the limits of integration tend to a limit. 4. more generally, of any function, approaching a limit. 5. (of a sequence or net in a topological space) eventually in every neighborhood of the point. See net convergence. See absolute convergence, conditional convergence, convergent in mean, convergent in measure. See also rate of convergence, linear convergence, uniform convergence, pointwise convergent. Compare divergent, oscillate.
| an - L | < ɛ for all n > N.
A corresponding definition applies in a metric space setting. 2. (of an infinite series) having a finite sum, generating a sequence of partial sums that has a finite limit. If the series a0 + a1 + a2 + ..., that is, the sequence of partial sums
〈a0, a0 + a1, a0 + a1 + a2,...〉,
converges, then the sequence 〈a0, a1, a2,... 〉 must converge to zero, but not necessarily conversely. For example, the sequence
〈1, 1/2, 1/3, 1/4,...〉
is convergent, but the series
1 + 1/2 + 1/3 + 1/4 + ...
is not. 3. (of an improper integral) having a finite value that is defined as the limit of the proper integrals as the limits of integration tend to a limit. 4. more generally, of any function, approaching a limit. 5. (of a sequence or net in a topological space) eventually in every neighborhood of the point. See net convergence. See absolute convergence, conditional convergence, convergent in mean, convergent in measure. See also rate of convergence, linear convergence, uniform convergence, pointwise convergent. Compare divergent, oscillate.