Theorem. The Integral Test Estimate. Suppose that

is a series which satisfies the hypotheses of the Integral Test using the function f and which converges to L. Let sn = a1 + a2 + ... + an be the nth partial sum and let rn = L - sn. Then

Discussion [Using Flash] [Using Java]

If we add sn to each term of the error estimate given in the theorem above, we obtain the following which provides a way to obtain a better estimate for value of the power series.