MATH 231 Lecture Notes - Lecture 17: Absolute Convergence, Ratio Test, Direct Comparison Test

• School: University of Illinois
• Department: Mathematics
• Course: MATH 231
• Professor: Bob Muncaster

Document Summary

We cannot use the previous convergence tests when a series does not consist of positive terms or if the series is not alternating. We can only apply the previous convergence tests to the absolute value of the original series. 1 n2 converges by the p test (p = 2, p > 1 ) 1 n2 converges and so by the comparison test we can say that n=1. Assume that an exists and equals l lim n an+1 if l < 1, then . An converges absolutely: of l > 1, or l = (the limit does not exist), then . 3) if l = 1, then no conclusion over absolute divergence can be made. These conditions apply to the value of the limit because when the values for n get very large: And we find that the relationship resembles a geometric series where l r. That is why, like with the value r, l<1 indicates convergence and l>1 indicates divergence.