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_i 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.

 

Divergence

If a sequence doesn't converge it is said to diverge. This can be if it goes to infinity, or if it simply cycles between two or more values without ever staying on one of them. For example, the sequences: 1, 2, 4, 8, 16, 32, ... and 1, 0, 1, 0, 1, 0, ... are both divergent. Although not immediately obvious, the harmonic series, 1 + 1/2 + 1/3 + 1/4 + ... (see harmonic sequence) is also divergent.