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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
s n is the sum of the area’s of these n rectangles. Math 9C Summer 2011 (UCR)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 16

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online