EDB — 1TX

Exercises

1. [1TX]Note:Dirichlet criterion for integrals.

   Let $f,g:[0,\infty )\to ℝ$ be continuous, where $f$ is positive and monotonic decreasing with $\underset{x\to \infty }{lim}f(x)=0$, while

   $$\underset{x>0}{sup}|{\int }_0^{x}g(t)\mathbb{d}t|<\infty .$$

   Then prove that

   $$\underset{x\to \infty }{lim}{\int }_0^{x}f(t)g(t)\mathbb{d}t$$

   converges.