In this article convergent sequences with natural number terms are investigated and the behaviors of tails and limits of these natural number sequences are explored. Firstly this study showed how the pre-service teachers response to the intuitive limit definition using "getting closer" for constant sequences. As a case of convergent natural sequences, the sequences in which the latter term is determined by the sum of digit squares of the former term are considered. To exploring these sequences the computational and charting capabilities of spreadsheets are utilized and some mathematical findings are obtained. Spreadsheet can be instrumentalized by teachers or students to provide a laboratory-like environment to explore a mathematical problem.