A class of infinite series is evaluated with the aid of Taylor's theorem and a comparison is made with other methods. In a recent note [1] a class of infinite series was shown to be equivalent to a number of definite integrals, and Taylor's theorem was used to establish convergence and to determine the sums of the series and the integrals to any desired accuracy. In this article a more general class of series to those in [1] are considered, and similar techniques are used to evaluate the corresponding sums and resulting integrals, some of which are improper. Alternative methods of evaluation include using upper and lower Riemann sums which give corresponding upper and lower bounds, as well as methods such as the trapezium rule. In the improper integral case it is not possible to use these, and so a transformation is used to make the integrals proper, leading to further interesting representations of the series as integrals, as well as the ability to evaluate the sums using a numerical method. Some specific worked examples are given using both Taylor's theorem and Riemann sums, and invite readers to consider other examples, as well as suggesting further areas for investigation. This material can be used as an illustration and reinforcement of a number of standard mathematical topics, and contains opportunities for student investigation.