Convergence is a property of some sequences and series. A sequence ui is said to be convergent if there exists a value u with the property that by choosing a large enough value of i, we can make ui as close as we wish to u. In other words, convergence is the tendency of a sequence toward a limit. In the case of a series, it is the tending toward of the consecutive partial sums of the series toward a limit. A sequence or series which does not converge is said to diverge.


For example, for the series


1 + (½) + (½)2 + ((½)3 + ...


the sum of the first two terms is 1.5, the first three 1.75, and the first four 1.875; as more and more terms are evaluated, the sum approaches the limiting value of (i.e., converges on) 2.