We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$. For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned}
r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\
S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\
R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t
\end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$. Show that, if $N$ is even: $N = 2 M$ with $M \geqslant 1$, and if $0 < x \leqslant 1 / N$, we have $S _ { N } ( x ) \geqslant 0$ and $$E _ { N } ( x ) \leqslant \frac { N ! x ^ { N + 1 } } { \sum _ { \ell = 0 } ^ { M - 1 } ( 1 - ( 2 \ell + 1 ) x ) ( 2 \ell ) ! x ^ { 2 \ell + 1 } }$$
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set
$$\begin{aligned}
r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\
S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\
R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t
\end{aligned}$$
The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Show that, if $N$ is even: $N = 2 M$ with $M \geqslant 1$, and if $0 < x \leqslant 1 / N$, we have $S _ { N } ( x ) \geqslant 0$ and
$$E _ { N } ( x ) \leqslant \frac { N ! x ^ { N + 1 } } { \sum _ { \ell = 0 } ^ { M - 1 } ( 1 - ( 2 \ell + 1 ) x ) ( 2 \ell ) ! x ^ { 2 \ell + 1 } }$$