1. tahirimanov19

    Infinite Continued Fractions

    I need help with ICFs, specifically calculating sum with algebra and/or simple calculus. $$A=a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + ....}}},$$ where $a_n = f(n), \; n \in \mathbb{Z}^+$. (?) Is the fraction above always convergent if $a_n \ge 1$? (?) What about $ 0 < a_n <...
  2. E

    continued fraction expansion of the complementary error function

    Could someone please explain to a non-mathematician why the continued fraction expansion of the complementary error function is the following: How does one come to this fraction as conclusion? I know and understand this term of the error function: I also know that erfc(z) = 1 - erf(z).
  3. W

    Understanding Continued Fraction Integer Factorization Method

    Please read it first. Thank you! Below is the table of the factorization process of number 11111. i qi pi Qi Ai-1 (mod n) 0 105 0 1 1 1 2 105 86 105 2 2 67 77 211 3 4 87 46 527 4 5 97 37 2319 5 2 88 91 1011 6 7 94 25 4341 7 1 81 182 9176 8 41 101 5 2406 9 3 104 59 7823 10 1 73...
  4. hyperbola

    Understanding projectile motion (continued)

    From my previous thread, with help from Mark FL and Skipjack: For a projectile object, it's maximum horizontal range (x_{max}) is twice it's maximum vertical range (y_{max}). x_{max} = \frac{v_R^2}{g} y_{max}= \frac{v_R^2}{2g} (x_{max} = 2y_{max}) In this instance, v_R is the...
  5. U

    Continued Fractions

    Determine the exact value of 2^0+2^0/(2^1+2^2/(2^3+2^6/....)
  6. D

    Continued fractions and irrationality

    I know how I can approximate irrational numbers using a continued fraction representation. Is it possible to do the reverse? That is, can I create an irrational number by coming up with some sort of continued fraction? Obviously I'd have to have some pattern or sequence that allows me to...
  7. R

    continued fractions help

    Greetings: I wish to prove that for every irrational number, r, there exists some simple continued fraction (CF) that converges to r. Suppose we know this to be true for every rational number. Is it sufficient to show that, for every interval containing r, there exists some number r’ within...
  8. E

    Etyucan's Fun Continued Fraction!

    Prove that [attachment=0:7j150ulb]gif.latex.gif[/attachment:7j150ulb]
  9. G

    continued fraction

    Hello, I was wondering if there is a way to easily convert a continued fraction to a simple fraction. so \dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\dfrac{1}{a_4+\dfrac{...}{a_n}}}}} to something like \frac{x}{y} so is there some formula to quickly calculate x and y for any n?
  10. M

    continued fraction expression for root 2 in Q_7

    Hensel's lemma implies that \sqrt{2}\in\mathbb{Q_7} find a continued fraction expression for \sqrt{2} in \mathbb{Q_7}
  11. mathbalarka

    Continued Fraction

    the continued fraction: [2,3,5,7,11,13,....] is convergent or not?
  12. E

    continued proportion

    I think this is one of the worst textbooks I'm using, but I recognize my limitations too. For instance, the authors tell me that a continued proportion is one in which a sequence of numbers as a, b, c, d is such that a/b = b/c = c/d which they also tell me is akin to the numbers 2, 6, 18, 54...
  13. A

    imaginary continued fraction

    what would be the value for \Huge\ F=\frac{1}{i+\frac{1}{i^2+\frac{1}{i^3+\frac{1}{i^4+...}}}} the same as: \Huge\ F=\frac{1}{i+\frac{1}{-1+\frac{1}{-i+\frac{1}{1+...}}}} noticing that (1,i,i^2,i^3,i^4,i^5,i^6,i^7,...)=(1,i,-1,-i,1,i,-1,-i,...)
  14. K

    Infinitely continued string of perfect squares

    All the positive perfect squares 1, 4, 9, 16, 25…. are written in accordance with strictly ascending order of magnitude and without the commas, resulting in the following infinite string: 149162536496481100121144........... Reading left to right, determine the 2010th digit in the...
  15. M

    find expression for a continued radical

    Hey guys, first off great forum.. I am posting for help with an extra credit problem. Find an expression for the continued radical: C = sqrt(m+sqrt(m+sqrt(m+... in terms of m that does not involve a continued radical. Then determine all positive integers m so that C is a positive integer...
  16. R

    Proper continued fractions

    Any real number can be written in the form a_1+1/(a_2+1/(a_3+1/...)) (where all the a_i's are integers) in exactly one way (except for trivial exceptions that involve the final a_i). I know that this is a finite series for rational numbers and an infinite series for irrational numbers, but is...