Here's the question you clicked on:
gerryliyana
find the taylor expansion around a singular point from a function f(z) = [exp(z)/(z-1)^{3}]!
find the taylor expansion around a singular point from a function \[f(z) = \frac{ \exp (z) }{ (z-1)^{3} } !\]
@oldrin.bataku @UnkleRhaukus @sirm3d @hartnn
Do you know what a singular point is?
@gerryliyana the point is that your radius of convergence is the distance your function is from its nearest singular point, where the function has a singularity (see: http://en.wikipedia.org/wiki/Mathematical_singularity#Complex_analysis ). I'm going to presume that stray \(!\) is a typo.
then, the question is wrong?
@gerryliyana no, but it won't converge I think.
with a radius of convergence of 0
ah ok.., then What form of expansion ??
Watch mate... we know \(\exp(z)\) is entire, but \(\frac1{(z-1)^3}\) is not -- it has a pole at \(z=1\). So we've found a singularity. Now let's try constructing a Taylor expansion around \(z=1\):$$\exp(z)=\sum_{n=0}^\infty\frac{e}{n!}(z-1)^n\\\frac{\exp(z)}{(z-1)^3}=\sum_{n=0}^\infty\frac{e}{n!}(z-1)^{n-3}$$oops, that's not a Taylor series, though -- it's a Laurent series.
yeah..., it's laurent.., i'm sorry, paper told us laurent., :). Thank you oldrin
@gerryliyana if you want it int he proper Laurent form rewrite:$$\frac{\exp(z)}{(z-1)^3}=\sum_{n=-3}^\infty\frac{e}{(n+3)!}(z-1)^n$$
ok.., thank you.., hei have a facebook ?? add me on fb at https://www.facebook.com/gerryliyana