Prove that the rational numbers [math]r\in\mathbb Q[/math] are exactly the real numbers whose decimal expansion is periodic.

Find geometric proofs, using triangles in the plane, for the well-known formulae for [math]\sin(x+y)[/math] and [math]\cos(x+y)[/math].

Develop some convergence theory for [math]x_n=a^n[/math] with [math]a \gt 0[/math], notably by proving that [math]a^n/n^k\to\infty[/math] for any [math]a \gt 1[/math], and any [math]k\in\mathbb N[/math].

Prove that [math]\sum_{n=0}^\infty\frac{1}{n!}=e[/math]. Also, prove that [math]\left(1+\frac{x}{n}\right)^n\to e^x[/math], and that [math]\sum_{n=0}^\infty\frac{x^n}{n!}=e^x[/math], for [math]x=-1[/math], then for [math]x\in\mathbb Z[/math], then for [math]x\in\mathbb R[/math].