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Re: Confusing Result with Series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg121268] Re: Confusing Result with Series
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Tue, 6 Sep 2011 04:00:08 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201109051105.HAA28332@smc.vnet.net>

Note that these are Puiseaux series. I think in the case of such series 
the "truncation order" in expression of the form  Series[f[x], {x, 0, 
n}] is not defined uniquely but depends on how the expansion is done. 
This is actually explained in the tutorial 
"MakingPowerSeriesExpansions":

The procedure that Series follows in constructing a power series is 
largely analogous to the procedure that N follows in constructing a 
real-number approximation. Both functions effectively start by replacing 
the smallest pieces of your expression by finite-order, or 
finite-precision, approximations, and then evaluating the resulting 
expression. If there are, for example, cancellations, this procedure may 
give a final result whose order or precision is less than the order or 
precision that you originally asked for. Like N, however, Series has 
some ability to retry its computations so as to get results to the order 
you ask for. In cases where it does not succeed, you can usually still 
get results to a particular order by asking for a higher order than you 
need.

However, there is another way to make sure you get a series with the 
specified truncation term. Just use this form f[x]+ O[x]^a  :

In[98]:= x/Sqrt[1+Sqrt[x]]+O[x]^(3/2)
Out[98]= x+O(x^(3/2))

In[99]:= Sqrt[x^2/(1+Sqrt[x])]+O[x]^(3/2)
Out[99]= x+O(x^(3/2))

In[100]:= x/Sqrt[1+Sqrt[x]]+O[x]^2
Out[100]= x-x^(3/2)/2+O(x^2)

In[101]:= Sqrt[x^2/(1+Sqrt[x])]+O[x]^2
Out[101]= x-x^(3/2)/2+O(x^2)

Andrzej Kozlowski


On 5 Sep 2011, at 13:05, 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
>





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