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Calculus and Beyond Homework Help
What is the Taylor expansion of x/sin(ax)?
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[QUOTE="RedDwarf, post: 6003140, member: 643451"] [B]Hey everyone 1. Homework Statement [/B] 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. [h2]Homework Equations[/h2] function: $$f(x) = \frac{x}{\sin (ax)}$$ Taylor expansion: $$ f(x) = \sum _{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ [h2]The Attempt at a Solution[/h2] 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. [/QUOTE]
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What is the Taylor expansion of x/sin(ax)?
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