Blog instead of Comments

So, since we really didn't do blogs over the holidays there isn't anything to comment on...so I will just review in a blog.

Taylor: they will give you c
f(c) + f^first deriv(c)(x) + f^second deriv(c)/2! (x-c)^2 + f^third dirv(c)/3! (x-c)^3

Maclauirn: is centered at 0
f(0) + f^first deriv(0)(x) + f^second deriv(0)/2! (x^2) + f^third dirv(0)/3! (x^3)

EXAMPLE:
Maclaurin up to the third degree

f=e^x
f(0)=e^0 = 1
f^1(0)= e^x(1) = 1
f^11(0) = e^x(1)(1) = 1

1+x+1/2! x^2
1+x+1/2(x^2)


Now don't forget about sequences and series! my fav :)
HERE ARE JUST A FEW THINGS TO REMEMBER ABOUT THE RULES:

p-series
1/n^p
p >1 CONVERGES
p <1 or =1 DIVERGES

geometric
(5/4)^n
n <1 CONVERGES
n >1 DIVERGES

limit comparison and direct comparison
*you must compare it to something easier
*use a different test
*then use this test to confirm first one

root and ratio
<1 CONVERGES
>1 or infinity DIVERGES
=1 INCONCLUSIVE

alternating series
*must take out the (-1)^n thing
*take limit MUST =0 or CANNOT be used


and something else we covered---

radius and interval of convergence:
*use the ratio test
*take the limit
*set up -1< x >1
*may have to solve

Post.

So we have a test. So here's a little review.

1. A sequence converges if it's limit is a number. It diverges if there's an infinity anywhere in it. For instance:
Given the sequence represented by the equation (n+1)/(n^2), say whether the sequence converges or diverges...at this point you would take the limit as n approaches infinity. In this case if would approach 0 because your limit rules say that if the degree of the top is less than the degree of the bottom, the limit approaches 0. Got it? So the entire sequence converges to 0 (a number)


2. Now for the difference between a sequence and a series. Indeed, I believe we learned this back in Advanced Math, but BRob stressed to us that Tir had issues with it, so might as well knock it in there a couple of times.

A sequence is just a list of numbers...aka...1, 3, 5, 7,...
**Note for this one it would be all odd numbers

A series is basically the same thing as a sequence, except that you have like addition signs in it...for example...3+4+5+6+7..
**Se those addition signs?? yeah, they're the ones you look out for..

3. Okay, so where I got a little tripped up was when we were saying: "If___, then___" But now, I think I've got it right..

Comment if you agree with this, "If the sequence of the series converges, then the series converges"

"If the sequence of the series diverges, then the series diverges"

My question is what are the conditions for ratio and root tests? I forgot those on the last quiz. :D

Post.

So we have a test. So here's a little review.

1. A sequence converges if it's limit is a number. It diverges if there's an infinity anywhere in it. For instance:
Given the sequence represented by the equation (n+1)/(n^2), say whether the sequence converges or diverges...at this point you would take the limit as n approaches infinity. In this case if would approach 0 because your limit rules say that if the degree of the top is less than the degree of the bottom, the limit approaches 0. Got it? So the entire sequence converges to 0 (a number)


2. Now for the difference between a sequence and a series. Indeed, I believe we learned this back in Advanced Math, but BRob stressed to us that Tir had issues with it, so might as well knock it in there a couple of times.

A sequence is just a list of numbers...aka...1, 3, 5, 7,...
**Note for this one it would be all odd numbers

A series is basically the same thing as a sequence, except that you have like addition signs in it...for example...3+4+5+6+7..
**Se those addition signs?? yeah, they're the ones you look out for..

3. Okay, so where I got a little tripped up was when we were saying: "If___, then___" But now, I think I've got it right..

Comment if you agree with this, "If the sequence of the series converges, then the series converges"

"If the sequence of the series diverges, then the series diverges"

My question is what are the conditions for ratio and root tests? I forgot those on the last quiz. :D

birfday blog :)

okkkk, so we had thanksgiving week off. and we had a takehome test. i shall do some problems from this test.

1. a sub n = (-2/3) ^ n

you plug in values starting at one, then two, three, etc...
& find your answer.

2. determine the convergence/divergence of the sequence with given nth term. if the sequence converges, find its limit.
a sub n = 3^n/5^n
geometric test. rewrite as (3/5) ^ n.
abs. value of r < 1 then it converges. 3/5 = r. after that, take the limit of the original problem. & it converges to that #.

these are just some examples. can someone explain integral test to me?

Combined late and now bloggg..

Alright..so i'll start off by saying these holidays went by entirely too fast..and obviously, you can't post blogs on iphones..because i actually did try in the airport going to new york..and it doesn't work...your keyboard does not pull up when you press in the blog box..odd. i know.

but anyway, lets get this thing started.

So, lets go over some throw back stuff..

1. Substitution: when the derivative is in the equation
ex: Scosxsinx
u = sinx
du = cos x
S u du
=1/2u^2 = 1/2sin^2(x) + c

2. By-Parts: when the derivative is not in the equation but you can sort of manipulate to figure it out...
ex: Sarctan(x)
u = arctan(x) dv = dx
du = 1/ 1+x^2 v = x
= arctan(x) (x) - S x/1+x^2
= arctan(x) (x) - 1/2ln(2)

3. Partial Fractions: when you have a fraction where the bottom can be broken up or factored
ex: umm..i can't think of one bc im not so good at these
But basically you have to break up the bottom and separate it to different fractions with differetn letters "naming" the fractions then you solve different systems to find your letters and then plug in to find the answer.

So, i've been working on my packet and i'm having alot of trouble remembering this stuff...and i'm getting kinda confused on some things..so come prepared on monday-wednesday..cause i'll have PLENTY questions for the test thursday :)