The classical definition of convergence of a sequence {s„} of real numbers may be extended by permitting the defining inequality to fail on an infinite, but relatively small,exceptional set of integers n. In this thesis the cases of exceptional sets of linear density zero and logarithmic density zero are considered. Basic properties of classical convergence are shown to hold for these cases, an example is constructed to show that a set of logarithmic density zero need not have linear density zero, and for each case a Tauberian condition sufficient to deduce classical convergence is provided
