Putnam Competition: Aftermath A?

Question

For those of you who are not aware, the Putnam Collegiate Mathematics Competition was held today, from 10:00-6:00PM US Eastern Time (So these are now fair game :) )

Here is one question that piqued my fancy, as I was unable to solve it in the time I had, given that it was stubbornly resilient to my class of usual approaches.  Anyone else?
No cheating by looking at the website.

A6)  Let f : R --> R be strictly decreasing with limit f(x) --> 0 as x --> infinity.  Prove that:

                 ∫_0^∞ (f(x)-f(x+1))/f(x) dx

diverges

gianlino · Accepted Answer

You can compare this to sum ( 1 - a_n+1 / a_n) for a decreasing sequence tending to 0.

Then ( 1 - a_n+1 / a_n) < ln [a_n /a_(n+1)], since for all x in R+  ln x <= x -1.

Since a_n ----> 0,  the sum  ln [a_n /a_(n+1)] diverges. You are done in the series case;

You can adapt it to the integral case.

Putnam Competition: Aftermath A?

1 réponse