this week we learned more about convergence, divergence, p-series, nth term, integral test, and we learned something new called direct comparison test.
i'll be completely honest, and say that i am really confused with a lot of this stuff. idk why it just isn't sticking in my head.
i need someone to go over pretty much all of it... especially direct comparison test.
i also don't know the difference. whenever brob says what kind is this. idk how to tell if it's p-series, or something else. i'm really lost. idk, this just wasn't my best week.
i know they all have to do with sigmas. and i know what to do if the steps are in front of me. but idk, i guess i'm just lost.
sorry if this is a short post, i'm just in need of help mostly.
i'll go over at least one thing.
INFINITY RULES:
top degree > bottom degree = +/- infinity.
top degree < bottom degree = 0
top degree = bottom degree = divide coefficients
these are actually very helpful and never go away. a lot of the time we just automatically do l'hospitals rule whenever we could just be doing this!
Sunday, October 17, 2010
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?
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?
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