2007-10-23

Today's nonsense
- It shall be written that Ketteman was singing choir, and was thus offtopic, and loud enough for Spiller to hear it. Therefore we can now do physics whenever.
- "Because his battery is out of camera." - Katie (in a stroke of what might be brilliance).

Direct Comparison Test

Please do not blame Spiller if there is an answer choice of "direct comparison test" on the test. So you should forget this stuff on the test, but then remember them after this next test. Direct comparison, we're going to have to do, because you have some simple series, and what can you tell me about this series?

Series from n=0 to infinity of 1/2^n. It converges because r=1/2 and the series therefore converges to 2. It's the first term (in this case, 1), divided by one minus r, because the first term is at n=0.
-- Series from n=0 to infinity of n/2^n not geometric because you have a variable in your numerator. If you have any variables that are not a part of your "r" then it's not geometric. How in the world do we do this? We compare it to something that we do know how to do.

Series from n=1 to infinity of 1/n^3. The sequence converges to zero. It's a p-series because p>1 in this case (p=3) and in this case n in the denominator will converge.
-- There's also series fromn=1 to infinity of 1/(n^3 + 1), and it's not a p-series. We did some integration like that before, and well, we can compare that to 1/n^3, so 1/n^3 converges and 1/(n^3 + 1) is less than it, and so we're going to have the Wall theorem all over again.

Your series are going to start to look more complex.

a_{_n} = n/(n^2 + 3)^2 is easy to integrate because it's u-substitution.

But ... b_{_n} = (n^2) / (n^2 + 3)^2 ... and what is the magical method of integration? It's not u-sub. It's The Magical Land of Trig Substitution.
-- 3(tan theta)^2.

Direct Comparison Test (for real?)

The Wall Theorem for Series

Let 0 < = a_{_n} < = b_{_n}, for all n,
If Series B_{_n} converges (because if the bigger one converges, then anything underneath it will converge it), then Series A_{_n} converges.
If Series A_{_n} diverges (because it's the smaller one, and anything above it will also diverge), then Series B_{_n} (the bigger one) diverges.

So, the box is going to go: if this series converges, then this series converges. Knowing that a_n is less than b_n, we can discuss the implications of the wall theorem.
- You could use this on the test tomorrow, but if it's an answer choice then don't choose it.

ex 1) Determine whether the series from n=1 to infinity of 1/(2+3^n) converges or diverges.
-- Ask whether or not 1/(3^n) converges or diverges. It converges to zero, and is bigger than the original, therefore the original converges as well.

Direct comparison test ... and then we will do the alternating series test next. And then the "great comparison test" (that's not what it's called). The ratio test is the way to go- the most powerful test, as you will find. Any questions on the test review?

Monotonic or nonmonotonic?
a_{_n} = (3n+1) / n^2
So is it always nonincreasing or is it always nondecreasing? Take the derivative and prove that the derivative is always positive or always negative, and then it would be monotonic. The algebraic way says that, you take the first four terms (4, 7/4, ) and that get smaller, so you think the terms are getting smaller, so you want to prove that the a_{_n} > = a_{_n+1} and so you try to show that it holds, so substitute it into the equation. This is the algebraic way of proving that it is monotonic. "n" is always going to be positive. If you cross multiply the comparison, you can write it out neatly to (3n+1)(n+1)^2 > = (3n+4)n^2. .... so let's look at the derivative version.