Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 2

Multiply the convergent Taylor series $\cos x=1-x^2/2+x^4/24+\cdots$ and $\sinh y=y+y^3/6+y^5/120+\cdots$. Keeping all terms through total degree five givesThere are six nonzero monomials displayed, so the answer does not select only part of the tied total-degree-five terms when asking for at least four terms. Both original series converge for every finite real $x,y$.

Use the Maclaurin series $\ln (1+x)=x-x^2/2+x^3/3+O(x^4)$ and the geometric series $(1+y{)}^{-1}=1-y+y^2-y^3+\cdots$. Multiplication, grouped by total degree, givesThe six displayed nonzero terms include every contribution through total degree three. A sufficient open region for the convergent product expansion is $|x|<1$, $|y|<1$.