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I have a tutorial question on Taylor series.

$$f(x,y,z) = e^{(\sqrt{x y}+\sqrt{x z}+\sqrt{z y})}$$

I managed to get the Taylor series at (1,1,1)

Is it possible to get the expansion at (0,0,0)? Because at (0,0,0) the derivative is 0/0 which is confusing...how do we evaluate 0/0 in Taylor series?

Thanks!

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$f$ is not differentiable at $0$, so there is no Taylor series... –  copper.hat Aug 30 '12 at 5:31
    
Would it be possible to make approximation for small x? –  Ken Ryou Aug 30 '12 at 5:42
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Well, it depends on what you mean, since $e^x \approx 1+x$ near $0$, you could approximate $f(x) \approx 1+\sqrt{x y}+\sqrt{x z}+\sqrt{z y}$. –  copper.hat Aug 30 '12 at 5:45
    
I just read about L hospital, would it be applicable in this case? Thanks! –  Ken Ryou Aug 30 '12 at 6:59
    
Not really, $f$ is really not differentiable at $0$. Loosely, the problem is that $x \mapsto \sqrt{x}$ has infinite slope at 0. (If it has a Taylor series, it would be differentiable; that's why $f$ has no Taylor series at $0$.) –  copper.hat Aug 30 '12 at 7:01
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1 Answer

up vote 2 down vote accepted

As copper.hat said in the comments, it is not possible to do a Taylor series in $x$, $y$ or $z$ around the point $(0,0,0)$ because the function has no derivative in that point.

What you can do in this point, however, is to do a Taylor expansion of just the exponential function, and insert the argument; this will give you an expansion in square roots: $$\begin{aligned} \mathrm e^{\sqrt{xy}+\sqrt{xz}+\sqrt{yz}} &= \sum_{n=0}^\infty \frac{1}{n!}(\sqrt{xy}+\sqrt{xz}+\sqrt{yz})^n\\ &= 1 + \sqrt{xy}+\sqrt{xz}+\sqrt{yz}\\ &\phantom{= 1} + \frac12 xy+\frac12 xz+\frac12 yz + \sqrt{xy}\sqrt{xz}+\sqrt{xy}\sqrt{yz}+\sqrt{xz}\sqrt{yz} + \dots \end{aligned}$$

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Hi Celtschk, Thanks. Is there any assumption in this expansion?Would this expansion be a valid one at (0,0,0)? –  Ken Ryou Aug 30 '12 at 7:16
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@KenRyou: There's no assumption here (that's why I didn't multiply out the square roots for simplification: that would invalidate the formula when not all arguments are positive; with complex square roots such manipulations are in general not allowed). The exponential function series has a convergence radius of infinity, therefore there's no restriction here. –  celtschk Aug 30 '12 at 7:34
    
Thanks for the explanation! –  Ken Ryou Aug 30 '12 at 7:50
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