- #1

idobido

- 10

- 0

- Homework Statement
- show that the given function is decreasing.

- Relevant Equations
- derivative

induction

As a follow up for : https://www.physicsforums.com/threa...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 ## k\in\left[3,\infty\right) ##

thing I've tried:

showing that first derivative is non-positive, but I've got complicated expression i cannot handle with (## k = x ##):

## \dfrac{\left(\frac{x-1}{x}\right)^{\ln\left(2\right)\,x}\,\left(\ln\left(2\right)\ln\left(\frac{x-1}{x}\right)\left(x-1\right)+\ln\left(2\right)\right)}{x-1}-\dfrac{\left(\frac{x-2}{x}\right)^{\ln\left(2\right)\,x}\,\left(\ln\left(2\right)\ln\left(\frac{x-2}{x}\right)\left(x-2\right)+2\ln\left(2\right)\right)}{x-2} ##

trying to compare

## \alpha\left(k\right)\ ## with ## \alpha\left(k+1\right)\ ##

also got complicated.

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 ## k\in\left[3,\infty\right) ##

thing I've tried:

showing that first derivative is non-positive, but I've got complicated expression i cannot handle with (## k = x ##):

## \dfrac{\left(\frac{x-1}{x}\right)^{\ln\left(2\right)\,x}\,\left(\ln\left(2\right)\ln\left(\frac{x-1}{x}\right)\left(x-1\right)+\ln\left(2\right)\right)}{x-1}-\dfrac{\left(\frac{x-2}{x}\right)^{\ln\left(2\right)\,x}\,\left(\ln\left(2\right)\ln\left(\frac{x-2}{x}\right)\left(x-2\right)+2\ln\left(2\right)\right)}{x-2} ##

trying to compare

## \alpha\left(k\right)\ ## with ## \alpha\left(k+1\right)\ ##

also got complicated.