Integral test for convergence or divergence

The integral test for convergence is only valid for series that are 1) Positive: all of the terms in the series are positive, 2) Decreasing: every term is less than the one before it, a_(n-1)> a_n, and 3) Continuous: the series is defined everywhere in its domain. The integral test tells us that, if the integral converges, then the series also converges. But if the integral diverges, then the series also diverges.

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How to find the error or remainder of a series

If you need to find the sum of a series, but you don’t have a formula that you can use to do it, you can instead add the first several terms, and then use the integral test to estimate the very small remainder made up by the rest of the infinite series. The sum of the series is usually the sum of the first several terms, plus a very smaller error that you can estimate.

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