Recent content by idobido

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    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    i observed i = k*ln(2) numerically, i cant use approximation, i need to show that exactly. trying to follow te comments here: https://math.stackexchange.com/questions/4748040/let-k-in-mathbbn-show-that-there-is-i-in-mathbbn-s-t-left1-frac1?noredirect=1#comment10074764_4748040 i got stuck at...
  2. I

    Show that the given function is decreasing

    did you notice that ## i really can't understand why the last result is true, you are applying ## \sqrt{k \cdot \log 2} ## on both expressions inside the derivative brackets i.e. (before the derivative)? why is that true that the inequality will hold after the derivative?
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    Show that the given function is decreasing

    how does it cancelled i've got ## \frac{d}{dk}\left( f(k)^k - g(k)^k \right) = \left( \ln f(k) + k \cdot \frac{1}{f(k)} \cdot f'(k) \right) \cdot f(k)^k - \left( \ln g(k) + k \cdot \frac{1}{g(k)} \cdot g'(k) \right) \cdot g(k)^k ##
  4. I

    Show that the given function is decreasing

    did you notice that ## g(k) = \left( \left(1 - \frac{1}{k}\right)^{\log 2} - \frac{1}{k} \right)^{\log 2} \neq \left(1 - \frac{2}{k}\right)^{\log 2} ## ?
  5. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    1
  6. I

    Show that the given function is decreasing

    As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/ show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
  7. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    ok, only need to show now that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for k>=3, i tried to show the first derivative non-positive but the expression i got is really complicated.
  8. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    already did, it seems the f(k, k*ln2) >=0, i.e. k*ln2 is the solution, but this needed to be shown analytically, and k*ln2 is not a natural number.
  9. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    for k = 1 or 2 the solution is trivial, i tried online calculator and it seems the result is about i = k*ln(2), but i can not show exactly how, the expression after setting i = k*ln(2), and his first derivative is very complicated to deal with...
  10. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    let ##k \in\mathbb{N},## Show that there is ##i\in\mathbb{N} ##s.t ##\ \left(1-\frac{1}{k}\right)^{i}-\left(1-\frac{2}{k}\right)^{i}\geq \frac{1}{4} ## I tried to use Bernoulli's inequality and related inequality for the left and right expression but i the expression smaller than 1/4 for any i...
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