How to calculate approximation $I = \int\_{-1}^{1} e^{-x^{2}} $ by developing integrant into power series, and how many terms of series is needed, so error is smaller than $\varepsilon = 10^{-3}$ ?

We can rewrite this $I = \int\_{-1}^{1} e^{-x^{2}} $ as

$I = \int\_{-1}^{1} \sum\_{n=0}^{\infty}\frac{(-x)^{2n}}{n!}$

and when we integrate it we get

$2\sum\_{n=0}^{\infty} \frac{1}{n!(2n+1)}$ .

How to calculate from here approximation and how many terms of series are needed, so error is smaller than $\varepsilon = 10^{-3}$ ?