Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$ so that $g ( k + 1 ) - g ( k ) = f ^ { \prime } ( k ) + R ( k )$. By applying to $g$ the Taylor formula with integral remainder at order $2 p$, show that there exists a real number $A$ such that for every $k \in \mathbb { N } ^ { * } , | R ( k ) | \leqslant A k ^ { - ( 2 p + \alpha ) }$.
Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$ so that $g ( k + 1 ) - g ( k ) = f ^ { \prime } ( k ) + R ( k )$.
By applying to $g$ the Taylor formula with integral remainder at order $2 p$, show that there exists a real number $A$ such that for every $k \in \mathbb { N } ^ { * } , | R ( k ) | \leqslant A k ^ { - ( 2 p + \alpha ) }$.