Analysis 1/Gemischte Satzabfrage/28/Aufgabe/Lösung

${\displaystyle {}q}$

${\displaystyle {}0\leq q<1}$

${\displaystyle {}k_{0}}$

${\displaystyle {}\vert {\frac {a_{k+1}}{a_{k}}}\vert \leq q\,}$

${\displaystyle {}k\geq k_{0}}$

${\displaystyle {}\sum _{k=0}^{\infty }a_{k}}$

${\displaystyle {}I=[a,b]\subseteq \mathbb {R} }$

${\displaystyle f\colon [a,b]\longrightarrow \mathbb {R} }$

${\displaystyle {}x\in [a,b]}$

${\displaystyle f(x)\geq f(x'){\text{ für alle }}x'\in [a,b].}$

${\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }$

Dann gilt

${\displaystyle \int _{a}^{b}f(t)g'(t)\,dt=fg|_{a}^{b}-\int _{a}^{b}f'(t)g(t)\,dt.}$