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How does $left(1+frac{x}{j}right)^{-1}left(1+frac{1}{j}right)^x=1+frac{x(x-1)}{2j^2}+Oleft(frac{1}{j^3}right)$

Mathematics Asked on December 23, 2021

How does $$left(1+frac{x}{j}right)^{-1}left(1+frac{1}{j}right)^x=1+frac{x(x-1)}{2j^2}+Oleft(frac{1}{j^3}right)$$
my attempt: It looks like maybe the taylor series of the numerator at 1 times the geometric of the denominator:

$(1-(frac{x}{j})+(frac{x}{j})^2dots)(1+(1+1/j)^x(ln(1+1/j))(x-1)$

which obviously isn’t the right way. or simply expand the numerator and the second term of the polynomial:
$(1-(frac{x}{j})+(frac{x}{j})^2dots)((1+1/j)((1+1/j)^{x-1})$
which doesn’t give the x-1 on the right.

One Answer

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: $$=1+C(x,1)frac{1}{j}-frac{x}{j}+left(frac{x}{j}right)^2-C(x,1)frac{1}{j}frac{x}{j}+C(x,2)frac{1}{j^2}+dots$$ $$=1+C(x,2)frac{1}{j^2}dots$$

Answered by user5389726598465 on December 23, 2021

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