March 27 - Comparison tests

Ultimately, I want you to be able to look at a given series and quickly identify whether it converges or diverges. There will always be some problems that you have to stop and think about, but most should be pretty clear once you're comfortable with the p-series test ideas, the geometric series ratio test and upcoming - a sort of "hierarchy" of functions.

We talked some today about the theory of how you could prove that since the series Sum(1/n^2) converges (p-series test, p = 2), then the series Sum(1/(n^2 + 1)) also converges. The "+1" in the denominator is unimportant for very large values of n so it should basically behave like Sum(1/n^2). Either comparison test - direct comparison (where we note that term by term, 1 / n^2 is greater than 1 / (n^2 + 1) so the sum of 1 / n^2 will also be greater) or limit comparison (take the limit as n -> ∞ of the ratio of the terms and get a positive (finite) value ) would work.

Further notes from today are below.