Post #8?

Well lets just get started..

Direct Comparison Test:
-This is dealing with a sigma.
-You have to find an easier one to compare it to.
-You will us either the nth term test, p-series thing, integral test, geometric thing.

EXAMPLE:
Say you have (sigma thing): 4^n/(5^n +3)
4^n/(5^n +3)-->compare to 4^n/5^n -->same thing as (4/5)^n
*this is geometric because it would be multiply by 4/5
*so by the rule for geometric 4/5 < 1 -->converges

P-Series:
-These are so easy.
-it is n^p
-if p > 1 -->converges
-if p < or = -->diverges

EXAMPLE:
1/n^2
*check to make sure it is n^p (which yes it is)
*p=2
*by the rule p > 1 -->converges

FEW THINGS TO SET STRAIGHT:

sequence: list of numbers
-converges if it has a limit
-diverges if it doesn't have a limit
-monotonic-terms always increasing/decreasing
-if bounded & monotonic-->converges
-if monotonic & not bounded-->diverges
-if bounded & not monotonic-->can be divergent

series: add/sub terms in a seq
-if sequence of partial sums converges-->series converges
-if sequence of partial sums diverges-->series diverges
-arithmetic series never converges
-geometric converges if absolute vale of r <1

* 1/infinity = 0
* 1/0 = infinity
* lim x->infinity of arctanx = pi/2

QUESTIONS FOR YOU TO COMMENT:

I have questions on the homework form this weekend about direct comparison test. What would you compare these to?

ln n/n+1
1/n!
e^-n^2 --> would you do something like 1/n^2?