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.
Read MoreThe convergence or divergence of the series depends on the value of L. The series converges absolutely if L<1, diverges if L>1 or if L is infinite, and is inconclusive if L=1. The root test is used most often when our series includes something raised to the nth power.
Read MoreThe comparison theorem for improper integrals allows you to draw a conclusion about the convergence or divergence of an improper integral, without actually evaluating the integral itself. The trick is finding a comparison series that is either less than the original series and diverging, or greater than the original series and converging.
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