Question

Find the first five non-zero terms of power series representation centered at x=0x=0 for the function below. f(x)=ln(5−x) f(x)=ln⁡(5−x)

Answer 1

If $f(x)=\ln(5-x)$, then

$f(0)=\ln5$

$f'(x)=-\dfrac1{5-x}=\dfrac1{x-5}\implies f'(0)=-\dfrac15$

$f''(x)=-\dfrac1{(x-5)^2}\implies f''(0)=-\dfrac1{25}$

$f'''(x)=\dfrac2{(x-5)^3}\implies f'''(0)=-\dfrac2{125}$

$f^{(4)}(x)=-\dfrac6{(x-5)^4}\implies f^{(4)}(0)=-\dfrac6{625}$

So the first five terms of the series expansion around $x=0$ are

$\ln5-\dfrac x5-\dfrac{x^2}{25}-\dfrac{2x^3}{125}-\dfrac{6x^4}{625}$