10/31: Integral Test

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