The coefficients [tex]c_n[/tex] are the same ones used in the Taylor series for [tex]f(x)[/tex] centered at [tex]x=0[/tex]. So
[tex]c_n=\dfrac{f^{(n)}(0)}{n!}[/tex]
We have
[tex]f(x)=\ln(3-x)\implies f(0)=\boxed{c_0=\ln3}[/tex]
[tex]f'(x)=\dfrac1{x-3}\implies f'(0)=\boxed{c_1=-\dfrac13}[/tex]
[tex]f''(x)=-\dfrac1{(x-3)^2}\implies f''(0)=-\dfrac19\implies\boxed{c_2=-\dfrac1{18}}[/tex]
[tex]f'''(x)=\dfrac2{(x-3)^3}\implies f'''(0)=-\dfrac2{27}\implies\boxed{c_3=-\dfrac1{81}}[/tex]
For [tex]n\ge1[/tex], we have the pattern
[tex]f^{(n)}(x)=(-1)^{n-1}\dfrac{(n-1)!}{(x-3)^n}\implies f^{(n)}(0)=-\dfrac{(n-1)!}{3^n}[/tex]
so that
[tex]c_n=-\dfrac{(n-1)!}{n!3^n}=-\dfrac1{n3^n}[/tex]
