How do you find the taylor series expansion of f(x) =x/(1+x) around x=0?

You know that $\frac{1}{1 - x} = 1 + x + {x}^{2} + {x}^{3} + \ldots$
so, $\frac{1}{1 + x} = 1 - x + {x}^{2} - {x}^{3} + \ldots .$
therefore, $\frac{x}{1 + x} = x - {x}^{2} + {x}^{3} - {x}^{4} + \ldots = {\sum}_{k = 0}^{\setminus} \infty {\left(- 1\right)}^{k} {x}^{k + 1}$