Cauchy product of two different series

Question

How to build the Cauchy product of the two series : $$ \exp(x) = \sum_{k=0}^{\infty}\frac{x^k}{k!} $$ and $$ \sin(x) = \sum_{k=0}^{\infty}\frac{(-1)^kx^{2k+1}}{(2k+1)!} $$. Thank you for your help.

Answer 1

Hints: The Cauchy product of the two series is simply the Taylor series of $e^x\sin x$.

Since $e^x\sin x = \text{Im}[e^xe^{ix}] = \text{Im}[e^{(1+i)x}]$, you can simply compute the Taylor series of $e^{(1+i)x}$ and take the imaginary part. This will be the same as the series for $e^x$, except $x$ is replaced by $(1+i)x$.

Using DeMoivre's Formula, $(1+i)^n = (\sqrt{2}e^{i\pi/4})^n = 2^{n/2}e^{in\pi/4}$