Presentation on theme: "Tests for Convergence, Pt. 2"— Presentation transcript:
1Tests for Convergence, Pt. 2 The Ratio TestorRelatives of the geometric series
2One Last TestThe Ratio Test: Suppose that is a series with positive terms and thatIf L < 1, then the series converges.If L > 1, then the series diverges.If L = 1, then the test is inconclusive---the series may converge or diverge.
3How Do We Apply the Ratio Test? Consider the series:Does it converge or diverge?What does this tell us?
4Why Does the Ratio Test Work? Answer: If the ratio test gives us an answer (it doesn’t always!), then the series is behaving “in the long run” like a geometric series.Recall the geometric seriesAnd this series converges when?
5Ratios of Adjacent Terms Convergence of series is all about “tail behavior”!One way to think about geometric series is to look at ratios of “adjacent” terms.All of these are equal to the same number, r.What about series whose adjacent terms are “almost” in a constant ratio “eventually.”
6Ratios of adjacent terms If we say that we are saying that for“large” values of n,In other words adjacent terms of our series are “almost” in a constant ratio “eventually.” So “in the long run” the partial sums should behave the same way as the partial sums of a geometric series.
7To be more specific. . .There is some point N in the sequence of terms so that for all n N,So we haveand so on. . .
8So our series is “almost” The “tail” is, basically, a geometric series!This should converge when?In the ratio test it says that we cannot draw any conclusions if L=1. Why would this be?
9Inconclusive. . . Consider the p-series What does the ratio test say about them?What conclusions can we draw? That is, what does it mean to say a test is “inconclusive”?