How does $$\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x=1+\frac{x(x-1)}{2j^2}+O\left(\frac{1}{j^3}\right)$$ my attempt: It looks like maybe the taylor series of the numerator at 1 times the geometric of the denominator:
$(1-(\frac{x}{j})+(\frac{x}{j})^2\dots)(1+(1+1/j)^x(\ln(1+1/j))(x-1)$
which obviously isn't the right way. or simply expand the numerator and the second term of the polynomial: $(1-(\frac{x}{j})+(\frac{x}{j})^2\dots)((1+1/j)((1+1/j)^{x-1})$ which doesn't give the x-1 on the right.
I saw my error right after posting. I'll leave this here in case someone else makes silly errors and searches while reading the same book: $$=1+C(x,1)\frac{1}{j}-\frac{x}{j}+\left(\frac{x}{j}\right)^2-C(x,1)\frac{1}{j}\frac{x}{j}+C(x,2)\frac{1}{j^2}+\dots$$ $$=1+C(x,2)\frac{1}{j^2}\dots$$