Let $n \in \mathbb { N }$ and let $f$ be a real function of class $C ^ { \infty }$ on $[ n , + \infty [$. We assume that $f$ and all its derivatives have constant sign on $[ n , + \infty [$ and tend to 0 as $+ \infty$. By applying, for $k \geqslant n$, the result of III.B.1 to $f _ { k } ( t ) = f ( k + t )$, show $$\sum _ { k = n } ^ { + \infty } f ^ { \prime } ( k ) = - f ( n ) + \frac { 1 } { 2 } f ^ { \prime } ( n ) - \sum _ { j = 1 } ^ { p } a _ { 2 j } f ^ { ( 2 j ) } ( n ) + \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$ where we have set $A _ { j } ^ { * } ( t ) = A _ { j } ( t - [ t ] )$ for every $t \in \mathbb { R }$. Show that $$\left| \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t \right| \leqslant \left| \frac { a _ { 2 p } } { 2 } \right| \left| f ^ { ( 2 p + 1 ) } ( n ) \right|$$
Let $n \in \mathbb { N }$ and let $f$ be a real function of class $C ^ { \infty }$ on $[ n , + \infty [$. We assume that $f$ and all its derivatives have constant sign on $[ n , + \infty [$ and tend to 0 as $+ \infty$.
By applying, for $k \geqslant n$, the result of III.B.1 to $f _ { k } ( t ) = f ( k + t )$, show
$$\sum _ { k = n } ^ { + \infty } f ^ { \prime } ( k ) = - f ( n ) + \frac { 1 } { 2 } f ^ { \prime } ( n ) - \sum _ { j = 1 } ^ { p } a _ { 2 j } f ^ { ( 2 j ) } ( n ) + \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$
where we have set $A _ { j } ^ { * } ( t ) = A _ { j } ( t - [ t ] )$ for every $t \in \mathbb { R }$.
Show that
$$\left| \int _ { n } ^ { + \infty } A _ { 2 p + 1 } ^ { * } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t \right| \leqslant \left| \frac { a _ { 2 p } } { 2 } \right| \left| f ^ { ( 2 p + 1 ) } ( n ) \right|$$