Presentation on theme: "Tests for Convergence, Pt. 2 The Ratio Test or Relatives of the geometric series."— Presentation transcript:
Tests for Convergence, Pt. 2 The Ratio Test or Relatives of the geometric series
One Last Test The Ratio Test: Suppose that is a series with positive terms and that If 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.
How Do We Apply the Ratio Test? Consider the series: What does this tell us? Does it converge or diverge?
Why Does the Ratio Test Work? Answer: If the ratio test gives us an answer (it doesnt always!), then the series is behaving in the long run like a geometric series. Recall the geometric series And this series converges when?
Ratios of Adjacent Terms 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. Convergence of series is all about tail behavior!
Ratios 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.
To be more specific... There is some point N in the sequence of terms so that for all n N, So we have and so on...
So our series is almost The tail is, basically, a geometric series! In the ratio test it says that we cannot draw any conclusions if L=1. Why would this be? This should converge when?
Inconclusive... Consider the p-series What conclusions can we draw? That is, what does it mean to say a test is inconclusive? What does the ratio test say about them?