By a hill function I mean a function such as $x^n/(K^n + x^n)$, where n is a real number smaller or greater than 1 up to around 10. Another way is to define a hill function is $\tanh((x/a)^n)$

I tried Expand on Wolfram|Alpha on x^1.2/(K^1.2+x^1.2) and it gave me an expansion of x about 1 (i.e., in terms of (x - 1)^i).

But in Mathematica I could not get the expansion.

Are there commands/packages other than Expand I can include for this purpose?

Series[x^1.2`/(K^1.2` + x^1.2`), {x, 1, 3}]

This expands your function about the point x=1 and gives 3 terms in the Taylor series.

In fact, if you type

expand x^1.2/(K^1.2+x^1.2) about 1

into Wolfram Alpha and then click the little button called "open code," it shows you something almost exactly like the Series expression above. Alternatively, you can type

= expand x^1.2/(K^1.2+x^1.2) about 1

into Mathematica. The = at the start of the line allows a much looser Wolfram-Alpha-like syntax. In this case, it again returns the same Series expression.

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