Dear all,
After trying days myself, I humbly ask for your advice with the following question:
Given that g(x,k)=x^k / alpha(x,k), where alpha(x,k)= Ln(x^(k+1)-1)) / (x^(k+1)-2))and x>= 2 and k>=1 and k is an integer (though this is not crucial), Show that g(x,k) is increasing in k (or equivalently: show that g(x,k+1)>g(x,k).
The following link contains a nicely formatted version of the question and my progress: https://www.dropbox.com/s/nn5voa8ukl8xwnk/q.pdf
The basic problem is the combination of the logarithm and the powers. Getting rid of the logs leaves me with a highly nonlinear equation, for which I only can find roots numerically (while I would like to solve the inequality analytically).
Thanks in advance,
Abe
After trying days myself, I humbly ask for your advice with the following question:
Given that g(x,k)=x^k / alpha(x,k), where alpha(x,k)= Ln(x^(k+1)-1)) / (x^(k+1)-2))and x>= 2 and k>=1 and k is an integer (though this is not crucial), Show that g(x,k) is increasing in k (or equivalently: show that g(x,k+1)>g(x,k).
The following link contains a nicely formatted version of the question and my progress: https://www.dropbox.com/s/nn5voa8ukl8xwnk/q.pdf
The basic problem is the combination of the logarithm and the powers. Getting rid of the logs leaves me with a highly nonlinear equation, for which I only can find roots numerically (while I would like to solve the inequality analytically).
Thanks in advance,
Abe