Get started today!

Good to have you back!
If you've signed in to StudyBlue with Facebook in the past, please do that again.

- StudyBlue
- Michigan
- Michigan State University
- Mathematics
- Mathematics 133
- Dong
- 10/31: Integral Test

Tiffany A.

10/31/08 11.3 The Integral Test Review If ∣r∣<1, = 2a) if is convergent, then b) If , then diverges 3. (p-test) 11.3 Theorem 1: let (an) be a decreasing sequence of positive terms. Let f be a continuous positive decreasing function such that f(n)=an. For n (some integer), then and , both either converge or diverge Ex 1. Harmonic series Show that =1+1/2+1/3+…1/n is divergent Solution: 1/x is continuous, positive, decreasing and f(n)=1/n=an diverges by the p test By the integral test diverges Example 2: Show that converges Solution: let f(x)=1/x2 1/x2 is continous, positive, decreasing And f(n)=1/n2=an dx converges by the p-test (p=2) Thus by the integral test, Example 3: Solution: set E, x1, x2 Example 4: show that converges Solution: left f(x) = Then f(x) is continuous, decreasing, positive dx Use direct comparison test, g(x)=1/x^2, since x^2+1>x^2, 1/x^2+1<1/x^2 dx converges by the p test Thus converges by the integral test Example 5: Show that diverges Solution: Let f(x)=1/xlnx Then f(x) is positive, decreasing, continuous for f(n)=1/nln2=an for n>2 dx == u=lnx, du=1/x Lnb-ln(ln2)=as b∞, lnlnb∞ dx diverges, thus diverges by the integral test

Advertisement

Advertisement

"StudyBlue is great for studying. I love the study guides, flashcards and quizzes. So extremely helpful for all of my classes!"

Alice , Arizona State University"I'm a student using StudyBlue, and I can 100% say that it helps me so much. Study materials for almost every subject in school are available in StudyBlue. It is so helpful for my education!"

Tim , University of Florida"StudyBlue provides way more features than other studying apps, and thus allows me to learn very quickly!Â I actually feel much more comfortable taking my exams after I study with this app. It's amazing!"

Jennifer , Rutgers University"I love flashcards but carrying around physical flashcards is cumbersome and simply outdated. StudyBlue is exactly what I was looking for!"

Justin , LSU