# Ngô Quốc Anh

## September 4, 2010

### CE: Convergence of improper integrals does not imply the integrands go to zero

Filed under: Counter-examples, Giải Tích 3 — Ngô Quốc Anh @ 12:31

This entry devotes a similar question that raises during a course of series. We all know that for a convergent series of (positive) real number

$\displaystyle \sum_{n=1}^\infty a_n$

it is necessary to have

$\displaystyle\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$.

This is the so-called $n$-th term test. A natural extension is the following question

Question. Suppose $f(x)$ is positive on $[0,\infty)$ and

$\displaystyle\int_0^{ + \infty } {f(x)dx}$

exists. Must $f(x)$ tend to zero as $x \to +\infty$?