post 8

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!

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?

post..

okay, so i'll start with things i don't understand.

1. i still don't understand how to tell if it converges/diverges. i skip all those problems

2. i don't understand improper integrals... they just don't click in my head.

3. i need help with some of those random trig function problems that are like S sec^6(2x). can someone briefly go over some rules about those.

4. trig sub... i can only do those on a good day. i'll get lost after finding x = ... (sqrt)x =..... and then i plug it in and i get stuck..

5. i'm also a litttleeeee confused on whether on not something is bounded. i didn't really get that. i understand monotonic though. which is what i'll go over

when it asks you if a sequence/series is monotonic... that means that it wants to know if it is always increasing/always decreasing.
so you plug in about 4 or 5 numbers (0-5), and see if it constantly goes up, constantly goes down, or isn't constant. if it is constant, it's monotonic. if it is NOT constant, it's NOT monotonic.
and i understand that usually your function is bounded by your first term... but i'm confused about the limit and stuff. you have to take the limit or something and sometimes it's bounded below? idk i'm kinda lost.

Post #7

A couple example problems that when I see I know exactly what to do.

1. S 3/(x-13)^6 dx

*basic substitution
u=x-13 du= 1
substitute: 8 S 1/u^6 = 8 S u^-6
integrate: (1/5)(8) S u^-5
plug in: -8/5(x-13)^-5 +C
which can be written as -8/5(x-13)^5 +C

2. S sinxcos^4x dx

*basic substitution b/c cos and sin are direct derivatives of each other
u=cosx du= -sinx
substitute: -S u^4
integrate: -1/5u^5
plug in: (-1/5)cos^5x +C
rewritten: -cos^5x/5 +C

3. S cos^3 6x dx

*break it up
break: cos^26x(cos6x)
identity: (1-sin^26x)cos6x
multiply in: S cos 6x - cos6xsin^26x
*substitute for each
u=6x u=sin 6x
du=6 du=6cos6x
1/6 S cosu - 1/6 S u^2
3(sin6x/6) - sin^36x/18
3sin6x/18 - sin^36x/18
sin6x(3-sin^26x)/18

4. lim 28-7x+4x^2/5x^2 -7
x-->infinity

*you may think L"Hopital's Rule BUT it is as x-->infinity so use your limit rules!
exponents equal each other so divide coefficients
=4/5

5. infinity S 2 3/x^5 (diverge or converge?)
Let me know if this is right!

a S 2 3x^-5 = -3(1/4)x^-4
3/4 x^-4
lim x--> infinty 3/4x^4
= 0 +3/64
so it = 3/64?

NOW MY BIG QUESTIONS:
How do you do improper integrals if its bounds are like 9 to 11, like no infinity?
How do you know you can use synthetic division on an integral?
Anyone know some tricks about trig sub?
How you do chasing the rabbit again?

Post...

Okay. So for the past couple of days we've been doing sequences and series. So, basically there are a few cardinal rules you should follow.. For example:

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"

I have a feeling that when the AP comes around, we'll most likely have to put this SOMEWHERE....anywho.

FOR STUFF YOU CAN COMMENT ON!

I truly was shaky on the whole find the sequence of the series...is that where you just find the terms plugged into the formula?

Also, I need help with sigmas...I'm used to having Step by Step Steps, and I don't. Could someone sum them up for me??? Thank you oh so much!

10/10/10 post

So since my last post, we have started to learn about sequences and series.

A sequence is a list of numbers defined by some equation, and a series is the addition or substraction of this list of numbers.

Converge vs. Diverge.
Sequences converge if they have a limit.
Sequence diverge if they don't have a limit.
If the limit of a sequence at infinity is infinity, then the sequence diverges.
If a sequence is bounded and monotonic* then it is converges.
If a sequence is bounded and not monotonic then the sequence diverges.
If a sequence is not bounded and monotonic then the sequence diverges.
*Monotonic - if terms are always increasing or always decreasing.
If a sequence of partial sums converge then the series converges.
If a sequence of partial sums diverge then the series diverges.

Series:
If something asks you to find the nth partial sum this means to find the sum at the nth term.
An arithmetic series will never converge. It will always diverge (as it approaches infinity).

Need to know:
Lim as n -> infinity (1 + (1/n))^n = e

Sequence properties follow limit properties (infinity limits at least).

Questions:

How do I use the squeeze theorem? I'm completely unsure of what to do.

How to find equations of series.

I'm somewhat unconfident of what to do anytime I see a sigma.

Ryan B.

No questions...

Yeah I realized I didn't post questions too! But I really do have questions. Maybe someone can answer these if they haven't done their comments..since no really did questions.

1. Like how you tell when it is divergent or convergent?

2. For improper integrals, sometimes they don't have an infinity so how do you break it up?

3. I need a reminder for how to do chasing the rabbit!

4. And how do you integrate 14x^27 cosx^14 dx?

No Questions.

Since there are no questions on anyone's blogs. I'll just do a mini-blog.

INTEGRATION:
Anytime there is an x term and an e term in a problem, use by-parts. The x term will always be your u and the e term will always be the dv.

Anytime there is an e term and a trig term, use by-parts until you see chasing the rabbit.

Anytime there is an x term and a trig term, use by-parts with your u as the x.

Questions:

S x^2 / x+3 I have no clue how to even start this.

This is from Calc. 1, but can anyone tell me the difference between a washer and (I don't know the other thing). THANKS

ryan

sorry

no one posted questions on their blogs. that's why i don't have comments. NEXT TIME, please post questions :)

Post...

Just some throwback steps straight from my old notebook. thought they MIGHT be useful to some...

First Derivative Test:
1. Take the derivative of the original problem.
2. Set the first derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the first derivative for x.
6. Solve.

Second Derivative Test:
1. Take the derivative of the first derivative.
2. Set the second derivative equal to Zero.
3. Solve for x.
4. Create intervals for x. i.e. (-∞, 1) (1, 4) (4, ∞)
5. Pick a number in the intervals then plug that number in the second derivative for x.
6. Solve.

limits:

Rule #1 - When the degree (exponent) of the bottom is GREATER than the degree of the top, the limit is Zero.
Rule #2 - When the degree (exponent) of the bottom is SMALLER than the degree of the top, the limit is infinity. (positive or negative)
Rule #3 - When the degrees are equal, the limit is the coeffecients.

linierazation:

The steps for solving linearization problems are:
1. Pick out the equation
2. f(x)+f`(x)dx
3. Figure out your dx
4. Figure out your x
5. Plug in everything you get

implicit derivatives:

First Derivative:
1. take the derivative of both sides
2. everytime you take the derivative of y note it with dy/dx or y^1
3. solve for dy/dx

Second Derivative:

first you find the first derivative and solve it for dy/dx by using the steps for the first derivative steps.
you then take the second derivative of the solved equation. Plugging in d^2y/d^2x everytime you take the derivative of y again. and where you have dy/dx you plug in your solved equation for that.
once you have everything plugged in and ready to go you then solve for d^2y/d^2x


HOW TO FIND THE EQUATION OF A TANGENT LINE:

1. take f'(x)
2. plug x in to find your slope m
3. plug x into f(x)to get y
4. using m and (x,y) plug it into the equation (y-y1)=m(x-x1).