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Re: Confusing Result with Series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg121247] Re: Confusing Result with Series
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 6 Sep 2011 03:56:17 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201109051105.HAA28332@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

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.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305




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