I found this interesting exercise on a calculus book (Stewart)
Let $ u=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\cdots $ $ v=x+\frac{x^4}{4!}+\frac{x^7}{7!}+\cdots $ $ w=\frac{x^2}{2!}+\frac{x^5}{5!}+\frac{x^8}{8!}+\cdots $ Show that $u^3+v^3+w^3-3uvw=1$ It turns out to be an interesting application of the 3rd root of unity, which greatly simplified the (could be) tedious calculation. I wonder if it has any deeper interpretation. (At least I don't see how to easily generalize it.) Can anybody explain this? (I don't need help on solving this problem.)