# Series expansion with bounded coefficients

Consider the two variable function $$f(x,y)=\sqrt{x^2+y^2}.$$ The function is continuous and therefore bounded on compact sets. We may expand the function as the following series using binomial series: $$f(x,y)=x+\frac{y}{2}-\frac{y^2}{8x}+\frac{y^3}{16x^2}-\cdots$$

Why is it that, even though, the function is bounded, the coefficients of the power series are unbounded in $$x$$? I guess this is due to the loss of differentiability at $$(0,0)$$. Is it possible to obtain a power series expansion for $$f$$ with bounded coefficients?