A simple convergent integral but not absolutely convergent.

Question

Anybody knows a simple example for convergent function but not absolutely convergent?

( simple = easy )

Thanks for coments!!!

Answer 1

Probably the simplest is $$\int_0^\infty \frac{\sin x}x\,dx.$$

Answer 2

Choose your favorite decreasing sequence $\{a_n\}$ of nonnegative numbers such that $\sum_{n = 1}^{\infty} a_n$ does not converge, and define the function

$$f(x) = \left\{\begin{array}{cc} a_1 :& 0 \le x < 1 \\-a_2:& 1 \le x < 2 \\ a_3 :& 2 \le x < 3 \\ -a_4 :& 3 \le x < 4 \\ \vdots \\ (-1)^{n + 1} a_n :& n-1 \le x < n \\ \vdots\end{array}\right.$$

Then it follows that

$$\int_0^{\infty} f(x) = a_1 -a_2 + a_3 - a_4 + \dots $$

converges, while

$$\int_0^{\infty} |f(x)| = a_1 + a_2 + a_3 + \dots $$

does not.