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Putnam Competition: Aftermath A?
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
1 Answer
- gianlinoLv 71 decade agoFavorite 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.
