2007-11-05

Today's nonsense
- Signed up the Math Club to the mailing list.
- Sean Rodgers
- Homework due tomorrow. We haven't talked about the pitiful root test.
- "No. We were in the valley. We were supposed to be roofing it but instead we were sticking around on the roof, so we shot nail guns at other roofs, and we could hear that the nails went far, you can ask Tyler Cromwell, they were up there with me, Aaron Mash, they were all down there working, we had to work, ..." - Ketteman's random story.
- Evidently we have notes from the Math Club. Thinking about a soup kitchen night, and taking the month of December off. Thinking about doing some boring activity while watching the movie on November 15th.
- Do the rabbits have tattoos in their ears and the pigs the tag? They squeal. You have to cut goat horns, they start bleeding, then they bleed for the whole day if you're name is Phoebe and you're freaking out. Spiller thinks that goats are psychotic.
-

The Pitiful Root Test so that we can make it an even number

Let Series a_{_n} be a series with nonzero terms.
1. Series a_{_n} converges absolutely if limit as n->infinity of (|a_{_n}|)^{^(1/n)} < 1

2. Series a_{_n} diverges if lim as n->infinity of (|a_{_(n)}|)^{^(1/n)} > 1.

3. The Root Test is inconclusive if
lim as n->infinity of (|a_{_(n)}|)^{^(1/n)} = 1

ex 1) Series from n=1 to infinity of e^{^(2n)} / n^{^(n)} === Series from n=1 to infinity of (e^{^(2)} / n )^{^(n)}
- So this test is all about figuring out what is being raised to what power. "This is the exact same as the geometric series test." - Dakota's revelation. Your "r" is now a specific value, and here it is much like an r, and if that thing much like an r is approaching a number less than one, you raise it to an nth root and it's already raised, so when you raise it to the nth root you're just taking away the n, so you're just looking at the inside part. So what we are asking in particular is:
limit as n->infinity of e^2 / n ... and 0 < 1 so it will converges absolutely.

Geometric series ... diverges if it is greater than or equal to 1. But in this case, of the root test, if it's already equal to 1 then the ratio test is inconclusive.

2) Series from n=1 to infinity of ( (n+1)/(2n+1) )^{^(n)} ...
- Goes to 1/2 .. converges. Geometric series test would point out that 1/2 is less than 1, so that's where geometric tests says it converges.