Determine the coefficient $a_{-4}$ in the Laurent expansion in the region $0<|z|<1$ of the function $f(z)=\frac{e^{1/z}}{1-z}$

I'm new using Laurent series so all help is appreciated.

My first thought was to write

$f(z)=\frac{e^{1/z}}{1-z}=\sum_{j=0}^\infty\frac{1}{n!z^n}\cdot \sum_{n=0}^\infty z^n$,

and continue from there.

Is this correct what to go? Or should i calculate otherwise?

  • $\begingroup$ it should be a z^j instead of z^n in your first sum (and you might want tu put parentheses), but other than that it's a good start. Now just expand the product, the coefficient you want will involve a sum over all $j$ and $n$ such that $j+n=-4$ $\endgroup$ – Glougloubarbaki Feb 15 '17 at 11:18
  • $\begingroup$ Thank you, that was really helpfull $\endgroup$ – Aerdennis Feb 15 '17 at 12:58

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