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.