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?


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