Hello community. I'm asked to show that if 1 < uk then (u-1/k)^n is greater or equal to u^n - (u^(n-1))*n/k. I'm supposed to use Bernoulli's inequality. I'd appreciate any hint. Please don't solve it for me
So given
$\displaystyle x > -1$ then $\displaystyle (1+x)^n \ge 1 + n x$
you want to show if
$\displaystyle uk > 1$ then $\displaystyle \left( u - \dfrac{1}{k}\right)^n \ge u^n - \dfrac{u^{n-1}n}{k}$.
Comparing the first part of the inequalities, have you thought of letting $\displaystyle x = \dfrac{-1}{uk}$?
I have found a counter-example, taking u = -7, k = -1/3 and n = 3. Apparently the author of the book from which I took the exercise meant 1 < u and 1 < k or maybe 1 < uk with the condition that both be positive. I had to go back to the text where he proves the existence of the positive nth root of c > 0, there he uses that inequality (which was giving me a headache already) in a crucial step but he's only working with positive numbers, then he asks one to prove it as an exercise later. So if we attach to standard notation, that meaning:"1 < uk" means the product of u and k is greater than one, he's basically telling you to prove something false.