Hello,
I need to find the maximum of a function [math]k_n = \frac{100!n}{(100-n)! \cdot 100^{n+1}}[/math]By quick empirical test I found out that it is increasing for few first n. So my first intuition was "what is the last increasing n" which I wrote as an inequality [math]k_{n} > k_{n+1}[/math]but unfortunately I was unable to solve it -> I got to the point [math]-100n^2 + 9999n + 99 > 0[/math] which just doesn't give sane solution. However if I turn my inequality to [math]k_{n+1} > k_{n}[/math] I get correct solution which is 9. So why does the first inequality fail to deliver the solution and what other methods could I use?
I need to find the maximum of a function [math]k_n = \frac{100!n}{(100-n)! \cdot 100^{n+1}}[/math]By quick empirical test I found out that it is increasing for few first n. So my first intuition was "what is the last increasing n" which I wrote as an inequality [math]k_{n} > k_{n+1}[/math]but unfortunately I was unable to solve it -> I got to the point [math]-100n^2 + 9999n + 99 > 0[/math] which just doesn't give sane solution. However if I turn my inequality to [math]k_{n+1} > k_{n}[/math] I get correct solution which is 9. So why does the first inequality fail to deliver the solution and what other methods could I use?