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[KaminaSquirtle]

Ok, I know what I need to do now, I just have no idea how to do it.  But at least it's progress.
Taking inspiration from this post:

I tried to take the easy way out and provide a physicist's answer.

Spoiler (click to show/hide)

Of course, I don't even have that. The stupid thing approaches infinity, and since my brilliant proof begins with "Assume that the series converges because otherwise what would be the point", I'm guessing I fucked up the java syntax. It's supposed to converge, isn't it? Oh well, no big loss. It wouldn't have been real proof, anyway.

I decided to confirm my suspicions about the answer using some short java code, to make sure I was on the right track.  As I expected, the answer is:

Spoiler (click to show/hide)

the square root of 2.

Of course the hard part is prove that this is the case.  But at least I know I'm on the right track.

[ILikePie]

Moved from a different thread:

we've been doing equations with variables I believe are called 'parameters' in English in class. The idea is that you have a variable that is to be treated like a normal number (so you get "x = 7 - a" as an answer, 'x' being the variable you need to solve, and 'a' being the parameter).
Anyway, I'm having trouble simplifying these equations. I have something like this "a(2x + 1) = 5(1 - x)", which I needs be simplified into something like "(a - something)x = something + a". Can someone please give a brief explanation on how to do this?

[KaminaSquirtle]

Spoiler: Solution (click to show/hide)

a(2x+1)=5(1-x)
Multiplying it out, we get:
2ax+a=5-5x
2ax+5x=5-a
(2a+5)x=5-a
If you also want solve for x:
x=(5-a)/(2a+5)
Just some simple algebraic manipulation.

And I believe the word is constants i.e. a is a constant.  That's how I've always heard them referred to.

[ILikePie]

Oh I see it now, I had trouble putting 2ax and 5x together. Thanks. It also is slightly more complex than just dividing everything by 2a+5, because you can't just go about dividing things by zero. It needs to be solved for when a = 2.5 (giving 0 = 2.5) and for when a = everything else (x =5-a/2a+5).
Thanks again.

[KaminaSquirtle]

Whoops, you're right. I'm getting careless about dividing by zero.  Must be all these sequences, where you only deal with positive integers.

If you're wondering about the case where a = -2.5, as you said we get 0=2.5.  This will be true no matter what x is.  So for a = -2.5, a(2x + 1) = 5(1 - x) for all x.

You're welcome.  It's refreshing to do some algebra after being foiled at every move by the problem I've been asking about previously in this thread.

[Vector]

KaminaSquirtle: do you have any special foods you eat for energy or something?  At this point, I'm stumped on a problem myself and working on it has been making my brain feel like it's made of butter.

Oh, and I've "lost" 4 pounds this week, so I suspect that I'm going to need to work on my caloric balance some as I get back into the studying groove.

[KaminaSquirtle]

KaminaSquirtle: do you have any special foods you eat for energy or something?  At this point, I'm stumped on a problem myself and working on it has been making my brain feel like it's made of butter.

Oh, and I've "lost" 4 pounds this week, so I suspect that I'm going to need to work on my caloric balance some as I get back into the studying groove.

Nope.  I just eat fairly healthy in general.  I think a lot of energy comes from getting enough sleep, though.  Maybe some kind of youthly vitality? 
I sure know what you mean about hard problems making your brain dissolve though.  I've had that happen to me plenty.  Haven't had it recently though.  My brain will probably feel like melted butter next semester though.  It'll be the first time I take a course higher than a 2xxx level.  It's a 4xxx course, "Introduction to Modern Algebra."  Description:
"Equivalence relations, greatest common divisor, prime decomposition, modular arithmetic, groups, rings, fields, Chinese remainder theorem, matrices over commutative rings, polynomials over fields."
I don't even know what most of those things are.  It'll be an interesting semester, if nothing else.

I'm willing to bet your problem is quite a bit harder than anything I'm dealing with too.  Sleep on it if you can?  That often helps me.  Or just take a break, do something mindless.  I know my brain, at least, can't work on stuff 24/7.

And I'm not going to work any more on that problem until my Mon., when my TA can answer some questions I have.  Not worth trying any more until I can clear some things up.

[Vector]

Nope.  I just eat fairly healthy in general.  I think a lot of energy comes from getting enough sleep, though.  Maybe some kind of youthly vitality? 

Nah, I'm just trying to break past the barriers I know I have, which come every two hours or so.  Last time I did this I ended up eating 5 meals a day; one time I ate a whole bag of almonds for dinner and needed more energy    I do eat healthily, but when I'm going for the daily 16-hour approach one needs to eat like an athlete.

That's what my professor told me, anyway, so I was wondering if you knew anything about it.

I know my brain, at least, can't work on stuff 24/7.

Mine has to, unfortunately.  Oh, well.  Mindless thing done.  Now off to get a bowl of rice and start work again.

[KaminaSquirtle]

That's what my professor told me, anyway, so I was wondering if you knew anything about it.

Nope, I honestly don't have anywhere near that kind of workload.

Mine has to, unfortunately.  Oh, well.  Mindless thing done.  Now off to get a bowl of rice and start work again.

:/

[Vector]

Ahh, the road to becoming a professional mathematician... I can feel the paranoia riding strong again

[Heron TSG]

Aye, the brain burns numerous calories from thinking so durn hard all the time. Some foods that might help with that are legumes, nuts, and fish. I've found fish to be helpful with The Calculus, so hopefully it has some effect on The Higher Maths.

[KaminaSquirtle]

Whoops, you're right. I'm getting careless about dividing by zero.  Must be all these sequences, where you only deal with positive integers.

If you're wondering about the case where a = -2.5, as you said we get 0=2.5.  This will be true no matter what x is.  So for a = -2.5, a(2x + 1) = 5(1 - x) for all x.

You're welcome.  It's refreshing to do some algebra after being foiled at every move by the problem I've been asking about previously in this thread.

God am I stupid.  0 never equals 2.5...  So it it false for all x.
Where is my brain?  I'm sure I put it somewhere around here...

[Vector]

Aye, the brain burns numerous calories from thinking so durn hard all the time. Some foods that might help with that are legumes, nuts, and fish. I've found fish to be helpful with The Calculus, so hopefully it has some effect on The Higher Maths.

Yeah, I computed it at ~8500 calories burned for 16 hours, plus basal metabolic rate.  No wonder I keep getting so hungry--and I'm nowhere near that level yet.

Can't eat fish, because I'm a vegetarian/close to vegan, but right now my diet consists mostly of beans, rice, nuts, and approximately a billion apples.  I have a fried egg and toast for breakfast, plus fruit, tea, and a bunch of water... might have to switch to two eggs, soon, especially because I'm going to have to bike to class next year.  Need to work on more green, red, and yellow vegetables (mostly squash right now), though I feel like I'm kind of eating pointlessly... every time I

Where is my brain?  I'm sure I put it somewhere around here...

This is how I feel.  I just typed up that I had gotten the right solution (whee!) and then it turned out that not only did I have the wrong solution, but that my solution I wrote out last year was pants-on-face retarded.

At least I can feel that the problem is an easy one.  It's getting there.


EDIT:

Spoiler: And for anyone who wants to share my pain: (click to show/hide)

Let S be a subset of T, and both equipped with a partial order relation.  Suppose that S is sup-dense in T.  Then prove that the canonical embedding is inf-continuous.

[Heron TSG]

Yeah, I computed it at ~8500 calories burned for 16 hours, plus basal metabolic rate.  No wonder I keep getting so hungry--and I'm nowhere near that level yet.

Sixteen hours? Are you mad? That's a lot of studying to do per day. Do you even have room to eat, sleep and take a single class?

I'm unsure as to whether or not the brain burns fat/muscle differently from physical exercise, but if they're the same, I can give advice on calories. I run for two to three hours a day, and must eat a lot to compensate for it. The thing is, the human body usually far prefers burning fat to muscle, because it wants to save the energy-rich muscle for when it really needs it. When you start burning nigh-10,000 calories a day, the body freaks out and you start burning muscle first. When your body burns muscle, you get tired rather quickly. Thus, protein and fats become extremely important, so that your body starts using fat before muscle again. As far as protein, nuts are good, but there are some interesting soy-based protein mixes that you can combine with milk, ice cream (assuming pseudo-veganism allows this) and some berries to produce some excellent milkshakes.

^{Protip: If you make protein-shakes, try to find a protein mixture that doesn't create clumps. Clumpy protein is horrid.}

[Vector]

Yeah, I computed it at ~8500 calories burned for 16 hours, plus basal metabolic rate.  No wonder I keep getting so hungry--and I'm nowhere near that level yet.

Sixteen hours? Are you mad? That's a lot of studying to do per day. Do you even have room to eat, sleep and take a single class?

Yup, it's fine.

As far as the brain goes, it likes using sugar--that's why my diet is so fruit-heavy.  Thanks for the advice.