This concept covers the Integral Test for infinite series, applied not only to determine convergence or divergence but to derive explicit bounds on the partial sums of a divergent series in terms of the corresponding improper integral, within single-variable calculus (theory of infinite series). Because the integrand is a decreasing function, comparing left-endpoint and right-endpoint rectangle sums to the area under the curve produces two-sided bounds relating the nth partial sum to the value of the integral evaluated at the same upper limit. This extends the qualitative convergence/divergence conclusion of the Integral Test into a quantitative estimate of the rate of growth (or decay) of partial sums, linking series behavior directly to integral evaluation.
This concept covers the Integral Test for infinite series, applied not only to determine convergence or divergence but to derive explicit bounds on the partial sums of a divergent series in terms of …