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 $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$. Show that $s$ is continuous on $[ 0,1 ]$. For continuity at 1, fix $\varepsilon > 0$ and show that if the natural integer $N$ satisfies $\left| r _ { n } \right| \leqslant \varepsilon$ for all $n \geqslant N$, then $\left| s ( x ) - s _ { N } ( x ) \right| \leqslant 2 \varepsilon$ for all $x \in [ 0,1 ]$. Then bound the modulus of $s ( x ) - s ( 1 ) = \left( s ( x ) - s _ { N } ( x ) \right) + \left( s _ { N } ( x ) - s _ { N } ( 1 ) \right) + \left( s _ { N } ( 1 ) - s ( 1 ) \right)$.
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 $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Show that $s$ is continuous on $[ 0,1 ]$.
For continuity at 1, fix $\varepsilon > 0$ and show that if the natural integer $N$ satisfies $\left| r _ { n } \right| \leqslant \varepsilon$ for all $n \geqslant N$, then $\left| s ( x ) - s _ { N } ( x ) \right| \leqslant 2 \varepsilon$ for all $x \in [ 0,1 ]$. Then bound the modulus of $s ( x ) - s ( 1 ) = \left( s ( x ) - s _ { N } ( x ) \right) + \left( s _ { N } ( x ) - s _ { N } ( 1 ) \right) + \left( s _ { N } ( 1 ) - s ( 1 ) \right)$.