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.)