Series expanding an expression to an arbitrary power

Question

How can I get Mathematica to directly produce the series expansion of an expression such as $(1+1/x)^n$, where $n$ is an arbitrary positive integer? Note that it's important in my expression to keep $n$ arbitrary rather than specifying a specific value.

When I enter something like,

 Series[(1 + x)^n, {x, 0, 5}]

it gives me the usual first few terms that the binomial expansion would produce but as soon as I enter the original expression instead, it just returns the expression itself. Of course for this particular case, I can just expand $(1+x)^n$ and take $x \rightarrow 1/x$, but I'm interested in expanding a more complicated expression which is a linear combination of expressions of the form

$$ \frac{(1+ax)^n(1+a/x)^n}{(1+x)(1+1/x)} $$

about $x=0$ and extracting the first few coefficients.

I've tried things like

Assuming[Element[n, Integers] && n >= 1, Series[f[x], {x, 0, 5}]]

but that doesn't help.

Answer 1

I'm not sure if I have understood exactly what you want.

But as it makes no sense to expand your expression about x=0, you can expand it about x=\[Infinity]:

Series[(1 + 1/x)^ n, {x, \[Infinity], 5}]

$1+\frac{n}{x}+\frac{(-1+n) n}{2 x^2}+\frac{(-2+n) (-1+n) n}{6 x^3}+\frac{(-3+n) (-2+n) (-1+n) n}{24 x^4}+\frac{(-4+n) (-3+n) (-2+n) (-1+n) n}{120 x^5}+O\left[\frac{1}{x}\right]^6$

Let's have a look at the other expression

f = ((1 + a x)^n (1 + a/x)^ n )/((1 + x) (1 + 1/x))

Series[f, {x, 0, 5}] // Simplify

$\left(\frac{a+x}{x}\right)^n \left(x+(-2+a n) x^2+\left(3-2 a n+\frac{1}{2} a^2 (-1+n) n\right) x^3+\left(-4+3 a n-a^2 (-1+n) n+\frac{1}{6} a^3 (-2+n) (-1+n) n\right) x^4+\left(5-4 a n+\frac{3}{2} a^2 (-1+n) n-\frac{1}{3} a^3 (-2+n) (-1+n) n+\frac{1}{24} a^4 (-3+n) (-2+n) (-1+n) n\right) x^5+O[x]^6\right.$

The first bracket is not expandable about x=0. Therefore it is left unchanged.

Summing up: just use Series about the apropriate Point.

Hope this helps, Regards Wolfgang