Padé approximant for a function $ f(x) $ always valid?

Question

Is there any function $f(x)$ which can not be approximated by a Padé rational approximant?

$$f(x) \approx \frac{a_0+a_1 x+ \ldots +a_nx^n}{b_0+b_1 x+ \ldots +b_m x^m} $$

What happens with $f(x)= \tan(x)$ or $f(x)=\log^{a}(x)$

Answer 1

Here is Padé approximation for $\tan(x)$

$$ \tan(x)\approx \frac{-{\frac {1}{135135}}\,{x}^{7}+{\frac {2}{715}}\,{x}^{5}-{ \frac {5}{39}}\,{x}^{3}+x }{ 1-{\frac {6}{13}}\,{x}^{2}+{ \frac {10}{429}}\,{x}^{4}-{\frac {4}{19305}}\,{x}^{6} } . $$