Calculating Sums Of Infinite Series

Most calculus textbooks leave the impression that the convergence or divergence of many infinite series∑∞n=1 an can be decided by appealing to appropriate tests, but except in special cases it is difficult to calculate the sum with precision, when the series converges.Leibniz's alternating series test provides a truncation error bound s ? sn < an+ 1 for a decreasing alternating series. Such an error bound yields an effective method of calculating the sum of the series with a given precision.Our purpose in this note is to show the proofs used to show convergence of positive series can be extended to give truncation error bounds and we can use the methods to calculate the sums of a few infinite series.