EDB — 1TX

Exercises

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

   Let \(f,g:[0,∞)→ℝ\) be continuous, where \(f\) is positive and monotonic decreasing with \(\lim _{x→∞} f(x)=0\), while

   \[ \sup _{x{\gt}0} |∫_ 0^ x g(t)\, {\mathbb {d}}t| {\lt}∞\quad . \]

   Then prove that

   \[ \lim _{x→∞} ∫_ 0^ x f(t) g(t)\, {\mathbb {d}}t \]

   converges.