- #1

RedDwarf

- 10

- 0

**Hey everyone**

1. Homework Statement

1. Homework Statement

I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here.

## Homework Equations

function: $$f(x) = \frac{x}{\sin (ax)}$$

Taylor expansion: $$ f(x) = \sum _{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$

## The Attempt at a Solution

I thought I could use the series form of sine:

$$sin(ax) = \sum (-1)^n \frac{(ax)^{2n+1}}{(2n+1)!} $$

$$x/sin(ax) = \sum (-1)^n \frac{ (2n+1)! } { a^{2n+1} }x^{-2n}$$

While this is in fact a series, this doesn't look like a Taylor expansion at all. Is there a clever way of seing the Taylor expansion without actually calculating all the derivatives by hand?

Wolfram Alpha gives a rather neat result, but I have no clue how one gets there.