MAT 21C Lecture Notes - Lecture 5: Ibm System P, Convergent Series, Integral Test For Convergence

• School: UC-Davis
• Department: Mathematics
• Course: MAT 21C
• Professor: Naoki Saito

Document Summary

X n=1 an converges, then an 0. X n=1 an diverges if lim n an fails to exist or is di erent from zero. Rule. p-series: p > 1: converges, p 1: diverges. Theorem 9. (the integral test) let {an} be a sequence of positive terms. Suppose that an = f (n), where f is a continuous, positive, decreasing function of x for all x n (n a positive integer). N f (x)dx both converge or both diverge. Caution: the integral test series an and the integral r . The integral is not the sum of the series. Theorem 10. (the comparison test) let p an,p cn, p dn be series with nonnegative terms. Suppose that for some integer n dn an cn. N > n. (a) if p cn converges then p an converges. (b) if p dn diverges then p dn diverges.