1. Tests for Convergence, Pt. 2
The Ratio Test or
Relatives of the geometric series

2. 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.

3. How Do We Apply the Ratio Test?
Consider the series:
Does it converge or diverge?
What does this tell us?

4. Why 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 series
And this series converges when?

5. Ratios 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.”

6. 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.

7. 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. . .

8. So 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?

9. Inconclusive. . . 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”?