We approximate $\varphi _ { n } ( x )$ using partial sums $$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$ Assume $N > p _ { 0 }$. Bound $\left| R _ { N } \right|$ as a function of $(N, n, x)$ with $$R _ { N } = \sum _ { p = N + 1 } ^ { + \infty } ( - 1 ) ^ { p } a _ { p }$$ Deduce, for fixed $\varepsilon > 0$, a sufficient condition on $N$ for $\left| \varphi _ { n } ( x ) - S _ { N } \right| < \varepsilon$.
We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Assume $N > p _ { 0 }$. Bound $\left| R _ { N } \right|$ as a function of $(N, n, x)$ with
$$R _ { N } = \sum _ { p = N + 1 } ^ { + \infty } ( - 1 ) ^ { p } a _ { p }$$
Deduce, for fixed $\varepsilon > 0$, a sufficient condition on $N$ for $\left| \varphi _ { n } ( x ) - S _ { N } \right| < \varepsilon$.