Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$. Let $x \in [ 0,1 ]$ and $n \in \mathbb { N } ^ { * }$. Show $$s ( x ) - s _ { n } ( x ) = r _ { n } x ^ { n + 1 } - \sum _ { k = n + 1 } ^ { + \infty } r _ { k } \left( x ^ { k } - x ^ { k + 1 } \right)$$
Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Let $x \in [ 0,1 ]$ and $n \in \mathbb { N } ^ { * }$. Show
$$s ( x ) - s _ { n } ( x ) = r _ { n } x ^ { n + 1 } - \sum _ { k = n + 1 } ^ { + \infty } r _ { k } \left( x ^ { k } - x ^ { k + 1 } \right)$$