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I have a question about the "Series" command. Specifically, the following input

Series[Sqrt[x^2], {x, 0, 2}]

gives me the following output

x+O[x]^3.

However, we know that this is not true for some $x$, e.g. for $x<0$. Because of the singular derivative at $x=0$, I would have expected Mathematica to give an error of some kind here. Moreover, when adding an assumption like this

Series[Sqrt[x^2], {x, 0, 2}, Assumptions -> (x < 0)],

I still get the same output 

x+O[x]^3,

whereas I know that the leading order should be $-x$ rather than $x$.

I am trying to understand why Mathematica gives me this result. Does "Series" somehow make some general assumptions about the variable of the expansion?

EDIT: I am using Mathematica version 10.2.

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    $\begingroup$ OK in Mma 11.1.0.0: the output under the assumption $x<0$ is $-x+O\left(x^3\right) $. $\endgroup$
    – user64494
    May 2, 2017 at 16:02
  • $\begingroup$ What version are you using? Recent versions appear to give sensible answers. $\endgroup$
    – mikado
    May 2, 2017 at 22:19
  • $\begingroup$ Edited the question, I am using version 10.2. Is this is a bug in this version? $\endgroup$
    – ScroogeMcDuck
    May 3, 2017 at 10:24

2 Answers 2

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This is an extended comment to demonstrate the effect of Assumptions in version 11.1.1 While a problem exists in some earlier versions, it has been corrected. 

$Version

"11.1.1 for Mac OS X x86 (64-bit) (April 18, 2017)"

Series[Sqrt[x^2], {x, 0, 2}]

enter image description here

Series[Sqrt[x^2], {x, 0, 2}, Assumptions -> (x > 0)]

enter image description here

Series[Sqrt[x^2], {x, 0, 2}, Assumptions -> (x < 0)]

enter image description here

The last two are special cases of

Series[Sqrt[x^2], {x, 0, 2}, Assumptions -> Element[x, Reals]]

enter image description here

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  • $\begingroup$ This seems more like an output I would have expected. I am using version 10.2, was this a bug in that version? $\endgroup$
    – ScroogeMcDuck
    May 3, 2017 at 10:27
  • $\begingroup$ Yes, this appears to be a bug in (some?) earlier versions (e.g., it gives the result shown for x > 0 in all four cases using v10.4.1 on my Mac). $\endgroup$
    – Bob Hanlon
    May 3, 2017 at 15:00
  • $\begingroup$ Cool, that's clear then. Could you add to your answer that there's this bug in some earlier versions? I'll accept the answer then since that answers my question $\endgroup$
    – ScroogeMcDuck
    May 8, 2017 at 13:04
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Bob Hanlon has shown which solution Mathematica 11.1.1 has determined. We can check it with the Taylor formula.

s = Sum[1/j! Nest[(x - x0)*# &, D[Sqrt[x^2], {x, j}] /. x -> x0, j], {j, 0, 2}]

((x - x0) x0)/Sqrt[x0^2] + Sqrt[x0^2]

Limit[s, x0 -> 0, Direction -> -1]
x

Limit[s, x0 -> 0, Direction -> 1]
-x

Mathematica is right.

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  • $\begingroup$ I am using version 10.2, which gave me the output in my question. Guess that version was bugged then? The command "Limit" does give the right result in 10.2 as well though. $\endgroup$
    – ScroogeMcDuck
    May 3, 2017 at 10:28

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