The discussion revolves around proving that a function f(x), which is greater than or equal to x for all x, must be a linear polynomial if the integral of 1/f(x) diverges as x approaches infinity. The context involves understanding the behavior of functions and their integrals in relation to divergence.
The discussion is active, with participants presenting differing viewpoints on the nature of f(x). Some participants have offered examples and counterexamples, indicating a productive exploration of the topic without reaching a consensus.
There is an ongoing examination of the assumptions regarding the polynomial nature of f(x) and its relationship to the divergence of the integral. The requirement that f(x) be greater than or equal to x is a key constraint under discussion.
HallsofIvy said:You can't, as stated, it is not true. There exist many non-polynomial functions that are "asymptotic" to x such that the integral of 1/f(x) diverges. However, if you require that f(x) be a polynomial, then it is true that f must be linear.
000 said:f(x) must always be larger than x, are there any asymptotic functions that fit that criteria?