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Re: Confusing Result with Series


In Mathematica 8.0.1, I'm seeing the same surprising result as you 
(albeit in ordinary OuputForm rather than the InputForm to which you 
converted the output) for

   Series[x/Sqrt[1 + Sqrt[x]], {x, 0, 1}]

namely, including the term -(1/2) x^(3/2), which certainly has order 
greater than the requested 1.

However, the result of

   Series[Sqrt[x^2/(1 + Sqrt[x])], {x, 0, 1}]

is essentially what you expected, namely, x + O[x]^(3/2).

Of course the two functions are NOT the same when, e.g., x < 0, although 
they are the same when x >= 0.

Still, the behavior with the first version of the function does seem 
contrary to the documentation. Perhaps somebody here will tell us why 
it's not.  Otherwise, I suggest you submit this directly to Wolfram 
Research as a bug.


On 9/5/11 7:05 AM, jschwab wrote:
> Hi Mathematica Gurus,
>
> I am seeing confusing behavior with the Series command, using
> Mathematica 7.0.1 on Mac OS X.
> Specifically, it is not always truncating at the order I would expect.
> Here's a a simple example of the problem I'm having.
>
> I expect that the result of a a series expansion of
> \frac{x}{\sqrt{1 + \sqrt{x}}}
> to first order will be
> x + O(x^{3/2})
> no matter how I write the input.
>
> Instead, I see the following behavior.
>
> In[175]:= Series[x/Sqrt[1 + Sqrt[x]], {x, 0, 1}]
> Out[175]= SeriesData[x, 0, {1, Rational[-1, 2]}, 2, 4, 2]
>
> In[176]:= Series[Sqrt[x^2/(1 + Sqrt[x])], {x, 0, 1}]
> Out[176]= SeriesData[x, 0, {1}, 2, 3, 2]
>
> Is this a known issue? Or some sort of expected behavior that I
> haven't understood?
> A naive Google search didn't reveal anything of particular relevance.
>
> Thanks,
> Josiah
>

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
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University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305




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