2007-10-22 - Review day
Today's nonsense
- Request to move the test back due to advisory period, Meg's baby shower, ..
- AIDS walk reference
- Avery's height, now officially at least 5'2''
- Avery's sandwhiches
- Magic land of the Trig == trigonometric substitution
- Test is now Wednesday/Thursday
- Camera battery charge is low
- Short shorts from decade day last Friday
#28 (Bobby) - See photograph
#3 (Phoebe) - She doesn't understand. See the photograph.
#4 - See the photograph. It converges because of the value of r, is less than one.
#5 - Phoebe - Area under the curve is integration.
#7 - Bobby - Determine which of the following converge. (A) diverges because p=1. (B) diverges because r>1 and it's geometric. (C) Diverges because p<1. (D) Diverges because p>1.
If the sequence converges to zero, you have to go on and try something else.
#21 - Phoebe - See photograph.
There is some integration by trig substitution. #26, #8, ... Bobby's #26.
#24 - Bobby - Determine if the following sequence converges/diverges. Sequence. Not series. So it's ((n+1)/n)^n. So we're just talking about the diverge. Which will it eventually converge or not? This is the whole "e" thing. It will probably be on the test.
#42 - Bobby - Determine the convergence or divergence of the series using the integral test: sum from n=1 to infinity of (1/n^2)cos(1/n)
This is u-substitution, and then you have -sin(1/n) from 1 to infinity, and if you plug in infinity, you get 0 - -sin(1). So if the integral converges, then the series converges. This is the integral test, or the "wall" thing that we do with our hands, it's similar to the wall, it's the integral version of the wall, it's the same idea except not.
#8 - Bobby - a trig problem (trig substitution)
Nth term test: take the limit as n->infinity, if you get a number anything other than 0, it diverges.
Some scribbling: what's on the review and what's on the test? If you're doing convergence/divergence of a series, now, the first thing you really want to look at, the nth term test, if the limit as n->infinity of your series does not equal zero, then the series is going to diverge. The nth term test determines your series. Sequence are the terms, series are the sums. Sequence means the pattern. What other things do we have to figure out divergence/convergence?
-- Geometric series test. How do you find the sum? It's the first term over 1-r. The absolute value of r has to be less than one in order to have a geometric sum work out here. So if r is greater than or equal to one, then it diverges, and there's no series to speak of.
-- Telescoping series. The telescoping series are the ones where you have to apply partial fractions. Because what you want to do is write out your first few terms and see what starts to cancel out. The ball bounce problem (2/3rds up every time, a geometric series, figure out the sum, double it, but then take away the first upwards distance because when you start to bounce it goes down once) is going to be on the test.
-- Integral test
-- P series (harmonic is just when it's 1/n or it's just when you integrate you get ln and so obviously with the harmonic it's going to diverge). This could even be 1/3n or 1/2n and obviously when you integrate this you are going to get a natural log. So more it's recognizing that it's a harmonic.
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#15 - Ketteman -
#34 - Bobby/Dakota - This is the same sequence that you need to know. See one of the other photographs.
#38 - Same thing. The main concept.
#29 - Bill - you can prove divergence with integral test. If there was an integral test and p series test choice, then you could choose either one, so since you can do one of them you have to mark one of the available choices, even though you can think of another. This means that you have to know each type of the tests.
Comparison test with sequences instead of series, that's what we will be doing tomorrow now instead of the test.