# convergence

Convergence is a property of some sequences and series.
A sequence *u _{i}* 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

*u*as close as we wish to

_{i}*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.