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Series and Sequences (yes, Calculus!)
urahonky
Cynical Old ManRegistered User regular
Cynical Old ManRegistered User regular
Hey guys,
A while back I posted a thread that really did help me get through the Calculus II final. You guys were great and I really appreciated the help! But now I'm stuck in yet another rut in Calc III. Series and Sequences.
I had a midterm two Wednesdays ago. I go the test back on the following Monday with a big fat 25 on the front of it. I nearly had a heart attack. I'd studied for about 2 days prior to the exam, and felt pretty comfortable with the exam when I turned it in. Obviously I need to stop feeling comfortable.
So she was nice enough to offer a retest, in which I took on Wednesday. I studied over the weekend, Monday, Tuesday, and called off work on Wednesday to study for the 6pm exam. I turned in the test with 5 blank out of 8.
So obviously that didn't work either. On TOP of that, I sent the instructor an email and come to find out... The final includes Series and Sequences. Just when I thought I might have been done, I'm going to have to deal with them yet AGAIN.
Whenever I get a problem in front of me... I can't figure out how to find out what it's asking for. I thought I may have finally figured it out, but it doesn't seem to be the case.
So I call on you fine folks to point me in the right direction to read through SOMETHING to help me out. I know that I should probably get a tutor, but my school doesn't seem to offer tutors over the Summer Quarter (I sent the email address two emails, but never got a response).
A while back I posted a thread that really did help me get through the Calculus II final. You guys were great and I really appreciated the help! But now I'm stuck in yet another rut in Calc III. Series and Sequences.
I had a midterm two Wednesdays ago. I go the test back on the following Monday with a big fat 25 on the front of it. I nearly had a heart attack. I'd studied for about 2 days prior to the exam, and felt pretty comfortable with the exam when I turned it in. Obviously I need to stop feeling comfortable.
So she was nice enough to offer a retest, in which I took on Wednesday. I studied over the weekend, Monday, Tuesday, and called off work on Wednesday to study for the 6pm exam. I turned in the test with 5 blank out of 8.
So obviously that didn't work either. On TOP of that, I sent the instructor an email and come to find out... The final includes Series and Sequences. Just when I thought I might have been done, I'm going to have to deal with them yet AGAIN.
Whenever I get a problem in front of me... I can't figure out how to find out what it's asking for. I thought I may have finally figured it out, but it doesn't seem to be the case.
So I call on you fine folks to point me in the right direction to read through SOMETHING to help me out. I know that I should probably get a tutor, but my school doesn't seem to offer tutors over the Summer Quarter (I sent the email address two emails, but never got a response).
urahonky on
0
Posts
Very sorry for the late reply. Crazy night.
Well... Honestly everything.
I think I've finally gotten the Ratio Test down, since that's pretty easy. But my problem is trying to figure out WHAT test I use, and how I get there.
For example (from what I remember on the exam), how do I solve:
n=1 -> infinity, (n^3)/3^n ? I stared at it for about 15 minutes.
Also I really hope I don't have to deal with this in my actual Programming field. I'd seriously change majors :P
Is that an example of a problem of determining convergence for a series or sequence? In any case I'd be inclined to use the ratio test because of the presence of the 3^n term will make it simplify nicely.
In any case, the only way to get this down is practicing until you want to stab yourself in the eye. Feel free to post your problems in the thread and I can walk you through these problems on a step by step process, explaining each step (I've been a TA for this subject for 3 semesters, so I know what I'm doing.) I might not be able to give you responses right away since I'm moving, but rest assured Ill get back to you eventually.
Also, this lets me practice with LaTeX, which produces mathematical text the way you'll see it in your textbook, so it'll be easy to read.
I'm such a fucking nerd..
Sequences and series is the worst part of calculus ever. EVER. Look at it this way: Calc 4 is a goddamn breeze compared to calc 3.
What you should do is spend a little time memorizing the indicators for which test of convergence should be used. If I recall correctly, there's five or six major tests. Just run through each of them for a given practice problem until you find the right test. Do this for enough problems, and it should start to become a sort of reflex.
The biggest problem I had with sequences and series wasn't that I couldn't solve the problem, it was that I didn't know which test to use until the exam was over.
Paul's Online Math Notes
That link specifically goes straight to the section on Sequences and Series. It says Calc II, but that's just how the classes work out at our university. He's a great professor in person, and his math notes are incredibly helpful for just about everything Calculus related.
In terms of general advice, I pretty much just have to repeat what the other guys here have said. The main difficulty with infinite series problems are figuring out which convergence test to use for a given problem. The only way to get good at them is to do about a million of them. When I was studying this I must have worked literally one or two hundred of these problems over the course of a few days to prepare for the test, and I'm glad I did. The only way to know which test to use is to work so many of them that you'll rarely encounter a type of problem you haven't already seen before. Eventually it'll click a little and you'll get good at it.
Oculus: TheBigDookie | XBL: Dook | NNID: BigDookie
Just looking at the problem, the ratio test does look appealing like Folken said because of the 3^n term. Whether or not it will work is still unknown, but it's probably the best one to start with. So, the ratio test says to define L = lim(n-->inf) |[An+1 / An]|, and based on L you can figure out whether the series converges absolutely, diverges, or if you need to use another test. L < 1 means it converges absolutely, L > 1 means it diverges, and L = 1 means this test doesn't tell you squat and you need to use another one.
In our case An = (n^3) / (3^n), so An+1 = [(n+1)^3] / [3^(n+1)].
This means L = lim(n-->inf) |[(n+1)^3] / [3^(n+1)] * [(3^n) / (n^3)]|.
And if I'm doing my algebra correctly, this simplifies to L = lim(n-->inf) [(n+1)^3] / 3(n^3).
Pull the 1/3 out, and the limit goes to 1, meaning L = 1/3. Boy I hope I did that right. There's a good chance I messed up somewhere since it's been so long since I did these, so check me on my math, but if I'm right, that of course means L < 1 and the series is absolutely convergent (and therefore convergent).
Often times the ratio test is good when you see a number raised to n, or if you see a factorial somewhere in the problem. In any case, I hope that helps. Sorry for the formatting, I hope you can read it and it makes sense.
Oculus: TheBigDookie | XBL: Dook | NNID: BigDookie
I'll have to go through that Paul's Online Math Notes thing again. I still have it bookmarked because of my last thread, but I didn't feel like reading through it because I thought I'd finally understood everything.
The final is on August 20th so I have time, and I'm not even sure how many series/sequences will be on a common final. Especially since we're learning vectors and stuff now (which is easy compared to last section). I just wish I hadn't BOMBED the hell out of the midterm (twice) so now it puts a lot of pressure on my final grade.
If you want to do something with Computer Graphics it doesn't hurt to pay attention to the Vector and Matrix stuff when it comes along though. They are used at the core of most (all?) 2D and 3D graphics libraries.
While the theory of programming involves little numbers, its application in some fields can. Depending on your interests or courses this might force you into more math than you'd like, but don't give up on programming because of that.
You are probably going to want to get familiar with exponentials (the 3^n is one) regardless of what you do in the short term, because they come up in a lot of things. Logarithms are essentially the inverse operation of an exponential and they have a base that is analogous to the 3 in the 3^n.
Even if you don't do anything mathy you want to be familiar with them, as they are a central aspect of finance, particularly interest. Being able to switch between different types of compounding interest and understanding how overwhelming their growth gets after time is necessary for a real good understanding of debt and investment.
There is a vector u, that goes from the point A(-6,0) to O(0,0), and v that goes from O(0,0) to B(2,-2). I am supposed to find a vector orthogonal to these two vectors.
So is it safe to assume the vector form is: u = <-6,0> v = <2,-2> or are am I doing something wrong? After reading through the notes I can't seem to find an example like this.
I went ahead and made a "new" vector called W, which was <a1,a2> and I was going to do the dot product between W and u, then W and v. Then solve for whatever I have, then put them into the form (a1)^2 + (a2)^2 = 1.
Does that make sense? But the problem is that I get 6a1 = 0 when I do the dot product of W * u, which means a1 = 0, which doesn't make sense so far from what I've been doing.
well, I don't think there is a vector that's orthogonal to both of those two vectors (at least a 2 dimensional vector); in two dimensions, an orthogonal vector in geometric terms is equivalent to perpendicular, and you can't have something perpendicular to two (non-parallel) vectors in 2-d... in general you can only have N orthogonal vectors for N dimensions.
In other words, your calculation is actually correct - any vector W = <0,a> for ANY value of a will be orthogonal to u; the vector <0,1> is the unit-length vector in that direction (your solving for 1, which is just to make the length of the vector equal to 1). However, none of these vectors will be orthogonal to <2,-2>. Are you sure that's how the question is worded?
There are other ways that I was taught to find orthogonal vectors, I think... but it's been a long time (someone else can answer, I have vague memory that there was a matrix-inverse-type way to do it)
Hmmm... Odd. I think I worded it correctly, but let me retype it exactly as it looks on the review sheet.
Let u be a vector from the point A:(-6,0) to O:(0,0), v be from the point O:(0,0) to B:(2,-2).
(a) Draw the vector 2u - v.
(b) Find a vector orthogonal to both u and v
(c) Evaluate u * v
Does that make sense?
A block of polyurethane with a mass of 100 grams is dragged up a frictionless incline of 30 (degrees) for a distance of 50cm. How much work is done by the gravitational force during this motion?
Okay, so W = |F||D|cos (theta)
You know that |D| = 50cm and cos(theta) = 30(degrees).
Would |F| = 100 (weight) * 9.8 (gravity)? Am I correct in thinking that?
This would make it: 42435.24N, which seems really high. But I can't see how else to do it.
as to your new post - Newtons is kg * m / s^2, so you need to convert your numbers to kg's / m's
.1 * 9.8 * .5 *cos(30)
(Also, bless you for answering at least. I like to brainstorm, but don't really have anyone to brainstorm with.)
I mean, unless you're measing your incline from the vertical, which I don't.
I didn't really read the question earlier (I was running out), but this is right, you want the height which would be sin(theta) * .50 (I'm assuming 50cm is the hypotenuse/total traveled distance - if the 50cm is the horizontal distance, then it would be tan(theta))
And a Newton is a fairly decent amount of force for a 0.1kg mass, so it won't be a big number