Continued Fractions of Functions 2

Determine the term numbers n (≤ 1000) during the expansion of the continued fraction and the result fn(x) in a table.

\(\normalsize Continued\ fraction\\ \hspace{50px} f(x)=b_0+{\large\frac{a_1}{b_1+{\large\frac{a_2}{b_2+...}}}}\\ (1)\ f(x)=\displaystyle \lim_{\small{n \to \infty}}f_n(x)\\ \hspace{50px}f_n(x)=b_0+{\large\frac{a_1}{b_1+}\frac{a_2}{b_2+}\ \cdots\ \frac{a_n}{b_n+}}\\ (2)\ Example\\ \hspace{20pt} function\hspace{20pt} b_0\hspace{38pt} a_n\hspace{22pt} b_n\\ \hspace{35pt} {\large\sqrt{x}} \hspace{28pt} 1\hspace{40pt} x-1\hspace{11pt} 2\\ \hspace{35pt} {\large\frac{x}{e^x-1}} \hspace{20pt} 1-{\large\frac{x}{2}}\hspace{20pt} {\large\frac{x^2}{4}}\hspace{20pt} 2n+1\\\)