# notes6 - Improper integrals and an introduction to the...

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Improper integrals and an introduction to the integral test So far, if we’re given an inﬁnite series and we want to know if converges or diverges there are only ﬁve diﬀerent things we can say: It’s geometric and converges, it’s geometric and diverges, it’s telescoping and converges, it’s telescoping and diverges or lim k →∞ a k 6 = 0 so it diverges. That’s it? So far, we don’t know very much! We want to build more tools to be able to tell more often when it converges. So the ﬁrst thing we do is remember what it means for an inﬁnite series to converge. An inﬁnite series converges if the sequence { s n } converges. Now we remember that one (but not the only) way to tell if a sequence converges is to ask if it’s bounded and monotone. Math 9C Summer 2011 (UCR) Pro-Notes June 26, 2011 1 / 16

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Now if each a k 0, then since s n = a 1 + a 2 + ··· + a n we know s n +1 - s n = a n +1 0 which, by one of our tests, means { s n } is increasing. So now if a k 0, the only thing left to check is that s n is bounded (from above) i.e., there is a number M such that s n < M for all n . Unfortunately, this is much more diﬃcult. One way to see if s n < M for all n is by thinking of each term a k in s n as the area of a rectangle with width 1 and height a k . Math 9C Summer 2011 (UCR) Pro-Notes June 26, 2011 2 / 16
s n is the sum of the area’s of these n rectangles. Math 9C Summer 2011 (UCR)

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## This note was uploaded on 02/22/2012 for the course MATH 9c taught by Professor C during the Fall '11 term at UC Riverside.

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notes6 - Improper integrals and an introduction to the...

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