The natural or asymptotic density of a strictly increasing sequence of positive integers is a general tool that can be used to determine if the corresponding sum of reciprocals converges and if so what the value of the sum is. Kempner's series is the subseries of the harmonic series formed by deleting all the terms that contain the digit nine. In this thesis, we explore the natural density of a strictly increasing sequence of positive integers. We also consider extensions of Kempner's series where the number of nines allowed is a function of the total number of digits a term has. We use natural density as a tool to determine under what conditions the new series converges and to estimate the value of the series.
