I am trying to find the taylor series of $f(x)=$ $\frac{1}{5+x'}$. And I cannot seem to get how to find the taylor series using the method I've been using for other functions. Another thing that's throwing me off is as to why in the question the $x'$ in the denominator has the apostrophe. I will also attach a picture to make sure I am reading it correctly. I will post my steps that I have taken so far, all help is appreciated. $$f(x)=\frac{1}{5+x'}$$ $$f'(x)=\frac{-1}{(5+x')^2}$$ $$f''(x)=\frac{2(5+x)}{(5+x')^{2\times2}}$$ $$f'''(x)=\frac{2(5+x)^4-2\times4(5+x)^4}{(5+x')^{2\times2\times2}}$$ $$f'''(x)=\frac{-24(5+x)^3}{(5+x')^{2\times2\times2\times2}}$$
I am seeing a pattern in the denominator, but none in the numerator and that $x'$ worries me. I will attach the exact question for reference. All help is appreciated, thanks.
I think the prime is a typo.
Write $$\frac{1}{5+x} =\frac{1}{(x+6)-1} =-\frac{1}{1-(x+6)} $$ Now this is the formula for the sum of a geometric series: $$\sum_{n=0}^{\infty}(-1)(x+6)^n $$ provided $|x+6|<1$.