Using the notation from II.B.2, where $\widetilde { S } _ { n , 2 p - 2 } ( \alpha ) = \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { k ^ { \alpha } } - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right)$, show that for every natural integer $p \geqslant 1$ $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p + 2 } ( \alpha )$$ and that $$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p - 2 } ( \alpha )$$ Deduce that the error $\left| S ( \alpha ) - \widetilde { S } _ { n , 2 p } ( \alpha ) \right|$ is bounded by $\left| a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) \right|$.
Using the notation from II.B.2, where $\widetilde { S } _ { n , 2 p - 2 } ( \alpha ) = \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { k ^ { \alpha } } - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right)$, show that for every natural integer $p \geqslant 1$
$$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p + 2 } ( \alpha )$$
and that
$$\widetilde { S } _ { n , 4 p } ( \alpha ) \leqslant S ( \alpha ) \leqslant \widetilde { S } _ { n , 4 p - 2 } ( \alpha )$$
Deduce that the error $\left| S ( \alpha ) - \widetilde { S } _ { n , 2 p } ( \alpha ) \right|$ is bounded by $\left| a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) \right|$.