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  1. J

    A Question about continued fraction representations

    How powerful are continued fraction representations? From what I understand, they could be used to exactly represent some irrational numbers So, could they represent any root of an nth degree polynomial equation? Specially where n>4, since 5th degree roots are not guaranteed to have an...
  2. J

    Proving inequalities with logarithm

    I think you're implying that I integrate both sides, but how do I choose the value of a and b?
  3. J

    Proving inequalities with logarithm

    log(b)-log(a) ?
  4. J

    Proving inequalities with logarithm

    log(x), but I still really can't where I'm supposed to be headed.
  5. J

    Proving inequalities with logarithm

    Can I have more clues please? I'm getting nowhere plus I'm not that good with integral calculus since we were never taught this in my high school & university.
  6. J

    Proving inequalities with logarithm

    I tried that, but to know if it is indeed monotonic I have to show that the selected expression in the term the picture I attached is always < 1 or > 1. So to proof of the original problem requires a proof for: n*(log(n+1)-log(n)) < 1 Right now it looks pretty circular.
  7. J

    Proving inequalities with logarithm

    I need to prove: (n+1)*(log(n+1)-log(n) > 1 for all n > 0. I have tried exponentiating it and I got ( (n+1)/n )^(n+1) < e. And from there I couldn't go any farther, but I do know that it is true by just looking at its graph. Could anybody help me please?
  8. J

    Implications of varying the definition of the derivative?

    Ohh, the function log(F(x)) ----------------------------- So this means that log(F*(x)) = F'( log(F(x)) ) right? What else does it imply or what else could it be applied to?
  9. J

    Implications of varying the definition of the derivative?

    If you take the logarithm, the denominator will be the same Δx, but the numerator will have logs wrapped around it log(F(x+Δx)) - log(F(x)) instead of F(x+Δx) - F(x).
  10. J

    Implications of varying the definition of the derivative?

    I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...
  11. J

    Why are linear equations usually written down as matrices?

    I've been taught that for any system of linear equations, it has a corresponding matrix. Why do people sometimes use systems of linear equations to describe something and other times matrices? Is it all just a way of writing things down faster or are there things you could do to matrices that...
  12. J

    Help in proving this inequality

    Given a new constrant that A+B = C+D = 1 Does showing that: d[ -1(a*log(a)+(1-a)*log(1-a)) ] / d[a] * d[ a*(1-a) ] / d[a] to be always greater than or equal to zero prove the original claim? Since satisfying this means that the two functions grow and shrink together (albeit not in the exact...
  13. J

    Help in proving this inequality

    The closest I've got is I've tried to log both sides of the 1st inequality giving log(a)+log(b) < log(c)+log(d) then I tried to make one side similar the the 2nd inequality but then I realized that I'm going in circles. How do I use the concave down point?
  14. J

    Help in proving this inequality

    Can somebody help me please, I've tried solving this for hours but I still couldn't get it. Given that a, b, c, d are positive integers and a+b=c+d. Prove that if a∗b < c∗d, then a∗log(a)+b∗log(b) > c∗log(c)+d∗log(d) How do I do it?
  15. J

    How to test if a sequence converges?

    Can you please explain ##\frac{|x_{k+1}-x_k|}{|x_k-x_{k-1}|}\leq a## more, I couldn't 100% get it
  16. J

    How to test if a sequence converges?

    I only used the 1/2 as an 'example answer' We know that it's not enough for the difference of 2 consecutive terms getting smaller is not enough, so by how much smaller does it have to be for it to converge?
  17. J

    How to test if a sequence converges?

    Given a sequence, how to check if it converges? Assume the sequence is monotonic but the formula that created the sequence is unknown. My first thought was if: seq(n+2) - seq(n+1) < seq(n+1) - seq(n) , is always true as n->infinity then it is convergent. Or in other words, if the difference...
  18. J

    How do I simplify this expression with a floor function?

    Unfortunately, I have no idea about what you said (I don't have that much of a background in maths, sorry) But I have been searching around to look for ways to solve this and I found that perturbation theory could solve algebraic problems that are not exactly solvable; would that work? I would...
  19. J

    How do I simplify this expression with a floor function?

    That's very clever; I think I would be able to use it. Thanks for the help! ------------ Edit: What I decided to do next is to solve for the minimum and maximum values of P using the 2 bounds. I'm having trouble with solving for P in the expression: \frac{(n+1)^{P}}{n^{P}+1}=X Is this even...
  20. J

    How do I simplify this expression with a floor function?

    I think the first step would actually be: R= ((n+1)^P + w) / (n^P + v) If it is, I have no idea how to go on from there..
  21. J

    How do I simplify this expression with a floor function?

    P is a real number greater than 0 n is a natural number (including 0) I'm trying to make a convergence acceleration formula for a (rounded up) sequence of numbers, then it eventually leads to this problem: seq(n) = ceil ((n+1)^P) / ceil (n^P) , for any real P>0 which requires me to completely...
  22. J

    How do I simplify this expression with a floor function?

    Regarding the 2nd option, I don't understand how to bring out P in the expression, ((n+1)^P + w1) / (n^P + w0)
  23. J

    How do I simplify this expression with a floor function?

    I would want to change this expression to another one that would have P easily accessible. \frac{\left \lceil (n+1)^{P} \right \rceil}{\left \lceil n^{P} \right \rceil} As an example of 'accessible'; given (n+1)^P / n^P, I can make P more accessible by using log(). Then it becomes P *...
  24. J

    How to solve this asymptotic equality?

    How do I solve for x in the relationship below: nx ~ n ln(n), as n -> infinity The answer that I'm getting is x=1, but that must be wrong since 1 ~ ln(n) as n-> infinity is wrong.
  25. J

    Which sequence acceleration method should I use?

    The sequence that I'm working with is sorta monotonic. It's monotonic most of the time, sometimes there's one number that ruins the trend like 0.0001001 in 0.1, 0.01, 0.001, 0.0001001, 0.0001, ... but those are very rare. Btw, the sequence above is just an example. Is Richardson extrapolation...
  26. J

    What does the Comp function mean?

    I think what it means is the general form of the any kind of primitive recursive function, but I'm not sure.. What do you think?
  27. J

    What makes these initial functions so special?

    Kleene said; http://plato.stanford.edu/entries/recursive-functions/#1.1
  28. J

    What makes these initial functions so special?

    People say that if you could break a function down into these three functions (constant, successor, projection or sometimes called initial/basic functions) using some operators, then it is primitive recursive. What makes these three functions so special?
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