Let $r$ be a non-zero natural integer and $f$ a function expandable as a power series on $] - 1,1 [$. We assume that for every $x$ in $] - 1,1 \left[ , f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$ and that $\sum _ { n \geqslant 0 } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$ converges absolutely. Show that $\int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } f ( t ) \mathrm { d } t = ( - 1 ) ^ { r - 1 } ( r - 1 ) ! \sum _ { n = 0 } ^ { + \infty } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$.
Let $r$ be a non-zero natural integer and $f$ a function expandable as a power series on $] - 1,1 [$. We assume that for every $x$ in $] - 1,1 \left[ , f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$ and that $\sum _ { n \geqslant 0 } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$ converges absolutely.
Show that $\int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } f ( t ) \mathrm { d } t = ( - 1 ) ^ { r - 1 } ( r - 1 ) ! \sum _ { n = 0 } ^ { + \infty } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$.