Cauchy product for power series of different powers

Question

How would you find the Cauchy product of two power series of different powers? For example, I want to find the Cauchy product of the two series $ \exp(x) = \sum_{k=0}^{\infty}\frac{x^k}{k!}$ and $ \cos(x) = \sum_{k=0}^{\infty}\frac{(-1)^kx^{2k}}{(2k)!}$ directly. I tried writing $x^{2k}$ as $(x^2)^k$, but I'm not sure if I can still use the Cauchy product definition in this form.

Answer 1

If you see the power series of the cosine function as$$\require{cancel}1+0\times x-\frac1{2!}x^2+0\times x^3+\frac1{4!}x^4+\cdots,$$then you have\begin{align}\exp(x)\cos(x)&=\left(1+x+\frac1{2!}x^2+\frac1{3!}x^3+\cdots\right)\left(1+0\times x-\frac1{2!}x^2+0\times x^3+\frac1{4!}x^4+\cdots\right)\\&=1+x+\cancel{\left(-\frac1{2!}+\frac1{2!}\right)}x^2+\left(-\frac1{2!}+\frac1{3!}\right)x^3+\cdots\\&=1+x-\frac13x^3-\frac16x^4+\cdots\end{align}As far as I know, there is no simple expression for this series.