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[Heron TSG]

Ah, forgot to factor out of both sides when I factored one of the numbers. Gotta stop doing that. Thanks for your help, I went back and fixed that, ended with this.

(x/(x+10))^1/2

I'm off to take a test in The Calculus! Factoring, you will not me this day! _{Even if I do every problem twice to make sure.}

[Vector]

But...

[Heron TSG]

But...?

What I mean is, I accidentally did something like this.

[(x²+2)+(x+3)] = x[(X+2)+(X+3)]

instead of this.

[(x²+2)+(x+3)] = x[(X+2)+(3)]

[Virex]

Not to burst your bubble, but
[(x²+2)+(x+3)] = x² + x + 5 =/=  x[(X+2)+(3)] = x² + 5x

[Jim Groovester]

But...?

What I mean is, I accidentally did something like this.

[(x²+2)+(x+3)] = x[(X+2)+(X+3)]

instead of this.

[(x²+2)+(x+3)] = x[(X+2)+(3)]

That is not how you factor.

If you're using the distributive property to pull out an x from a quantity, you have to do it to every term in the bracket. You did not do this.

Using your example,

[(x²+2)+(x+3)]

The proper way to pull an x out of this is the following:

[(x² + 2) + (x + 3)] = x[((x + (2/x)) + (1 + (3/x)))]

Do you see the difference? Every term. (Also, pulling out an x from an x leaves you with 1, not 0 like what you have.)

[Virex]

Made another update to my solution to the converging sequence problem. Hopefully now the proof is sound. I'd really like for someone to run it trough and see if it's not a load of bogus, but I guess I'll have to wait till Vector's done with her answer.

[KaminaSquirtle]

I'd love to, however I'm nowhere near solving it on my own, so I can't.  Right now I'm running through some simpler stuff, like proving Newton's Method, etc.  I'd like to ask two questions though, just to see if I have enough prerequisite knowledge right now.

1) Did you need to use anything involving Cauchy Sequences?  I think we get to those next week.

2) Is there more to it than proving it is monotonic for all n>=n₀, for some n₀ in the positive integers, and showing that it is bounded?  Because we haven't gone over any other ways to solve recursive sequences.  Like I said, they were only introduced this week.

Of course, this is assuming it converges, but I'm fairly sure it does.  Please don't tell me either way though, I really want to get this one on my own.

[Virex]

I'd love to, however I'm nowhere near solving it on my own, so I can't.  Right now I'm running through some simpler stuff, like proving Newton's Method, etc.  I'd like to ask two questions though, just to see if I have enough prerequisite knowledge right now.

1) Did you need to use anything involving Cauchy Sequences?  I think we get to those next week.

I did not explicitly use anything related to Cauchy sequences. I also can't tell you if it's a Cauchy function...

2) Is there more to it than proving it is monotonic for all n>=n₀, for some n₀ in the positive integers, and showing that it is bounded?  Because we haven't gone over any other ways to solve recursive sequences.  Like I said, they were only introduced this week.

That could work. I used a method similar to that, but not quite the same, because you can reduce the function to a much simpler form with the appropriate method

Spoiler: Don't read unless you want a small tip (click to show/hide)

(have you dealt with limits yet?)

[KaminaSquirtle]

I did not explicitly use anything related to Cauchy sequences. I also can't tell you if it's a Cauchy function, due to the method I used.

Excellent, I just wanted to make sure that I didn't need to know any of that.  Vector mentioning them worried me.

2) Is there more to it than proving it is monotonic for all n>=n₀, for some n₀ in the positive integers, and showing that it is bounded?  Because we haven't gone over any other ways to solve recursive sequences.  Like I said, they were only introduced this week.
That could work. I used a method similar to that, but not quite the same, because you can reduce the function to a much simpler form with the appropriate method

Spoiler: Don't read unless you want a small tip (click to show/hide)

(have you dealt with limits yet?)

Spoiler (click to show/hide)

Well yeah, that's kind of the point, to find the limit as n->infinity.  We've worked from the definition, with the limit theorems (If lim(s_n)=L₁, and lim(t_n)=L₂, then lim(s_n+t_n)=L₁+L₂, etc.), and with L'Hopital's Rule.  So yeah, I know a good amount about limits, this course is a good way past Calc 1, after all.
I will have to think more amount limits though.

[Virex]

I did not explicitly use anything related to Cauchy sequences. I also can't tell you if it's a Cauchy function, due to the method I used.

Excellent, I just wanted to make sure that I didn't need to know any of that.  Vector mentioning them worried me.

2) Is there more to it than proving it is monotonic for all n>=n₀, for some n₀ in the positive integers, and showing that it is bounded?  Because we haven't gone over any other ways to solve recursive sequences.  Like I said, they were only introduced this week.
That could work. I used a method similar to that, but not quite the same, because you can reduce the function to a much simpler form with the appropriate method

Spoiler: Don't read unless you want a small tip (click to show/hide)

(have you dealt with limits yet?)

Spoiler (click to show/hide)

Well yeah, that's kind of the point, to find the limit as n->infinity.  We've worked from the definition, with the limit theorems (If lim(s_n)=L₁, and lim(t_n)=L₂, then lim(s_n+t_n)=L₁+L₂, etc.), and with L'Hopital's Rule.  So yeah, I know a good amount about limits, this course is a good way past Calc 1, after all.
I will have to think more amount limits though.

Spoiler (click to show/hide)

Tip: you should end up with a function that doesn't depend on n

[KaminaSquirtle]

I did not explicitly use anything related to Cauchy sequences. I also can't tell you if it's a Cauchy function, due to the method I used.

Excellent, I just wanted to make sure that I didn't need to know any of that.  Vector mentioning them worried me.

2) Is there more to it than proving it is monotonic for all n>=n₀, for some n₀ in the positive integers, and showing that it is bounded?  Because we haven't gone over any other ways to solve recursive sequences.  Like I said, they were only introduced this week.
That could work. I used a method similar to that, but not quite the same, because you can reduce the function to a much simpler form with the appropriate method

Spoiler: Don't read unless you want a small tip (click to show/hide)

(have you dealt with limits yet?)

Spoiler (click to show/hide)

Well yeah, that's kind of the point, to find the limit as n->infinity.  We've worked from the definition, with the limit theorems (If lim(s_n)=L₁, and lim(t_n)=L₂, then lim(s_n+t_n)=L₁+L₂, etc.), and with L'Hopital's Rule.  So yeah, I know a good amount about limits, this course is a good way past Calc 1, after all.
I will have to think more amount limits though.

Spoiler (click to show/hide)

Tip: you should end up with a function that doesn't depend on n

Thank you, that should be enough for me to crack this thing.  Hopefully.  Anyway, that's enough tips for now, I feel I've allowed myself enough outside help.

[Vector]

Virex,

Spoiler (click to show/hide)

I don't know how good I feel about that initial limit-taking of n to infinity, and then playing with the equation's terms as though you hadn't done that.  Can you explain why that works?

Sorry, I really need to brush up on limits ^_^;

[Virex]

Virex,

Spoiler (click to show/hide)

I don't know how good I feel about that initial limit-taking of n to infinity, and then playing with the equation's terms as though you hadn't done that.  Can you explain why that works?

Sorry, I really need to brush up on limits ^_^;

Spoiler (click to show/hide)

 To be honest, you got me there. I assumed it works, since the function is defined for all positive integer values of n, meaning that even if you take the limit, the function should still retain it's behavior. In other words, for sufficiently large values of n, the behavior of the function approaches the given behavior and thus I assumed that at the limit, the function retains that behavior. The problem is of course that to do so, one first has to assume that the function behaves as stated at the limit...

[Vector]

Spoiler (click to show/hide)

 To be honest, you got me there. I assumed it works, since the function is defined for all positive integer values of n, meaning that even if you take the limit, the function should still retain it's behavior. In other words, for sufficiently large values of n, the behavior of the function approaches the given behavior and thus I assumed that at the limit, the function retains that behavior. The problem is of course that to do so, one first has to assume that the function behaves as stated at the limit...

Spoiler (click to show/hide)

I can't fully understand you (I'm completely exhausted), but really that doesn't seem logical to me--since what you've done is taken the limit of one part of the function without taking the limit of the entire thing...

I looked it up, and you have to take the limit of the a-n at the same time as you take the limit of the n, since they're two sequences multiplied by each other.  Don't know if that's what you were doing, but your notation doesn't express that as far as I can tell.

I think you're definitely on the right track, though.  It mostly looks right, just possibly a bit holey in some places.

[Heron TSG]

That is not how you factor.

If you're using the distributive property to pull out an x from a quantity, you have to do it to every term in the bracket. You did not do this.

Excuse me, I meant to say this.

[ (x²(2)) +  (x(3)) ]

becomes

x [ (x(2)) +  (3) ]

Durn multiplication/addition confusion. The simplest things are often the most easily missed.