I'm aware that there is only one power series representation for a function about a given center; however, based on graphical evidence, it seems that both of these series converge to the sine function. $$ \sin x = \sum_{k=0}^{\infty}\frac{(-1)^{k+1}x^{2k-1}}{(2k-1)!} = \underbrace{\sum_{k=0}^{\infty}\frac{(-1)^kx^{2k+1}}{(2k+1)!}}_{\text{standard textbook def}}$$ May someone clue me in to what I might be missing here?
Solved: The first series is not a power series.