Notes

We wish to show that for small $x$, $$\sqrt{1+x}\approx 1+\frac{x}{2}$$ To derive this formula, we use the well known fact that $$(1+x{)}^{\alpha }=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}\prod _{k=0}^{n-1}(\alpha -k)$$ The result follows by substituting $\alpha =1/2$ and truncating the series after two terms.