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Bug introduced in 9.0 or earlier and persisting through 12.2 or later

Executing the following

SeriesCoefficient[Log[1/2 (1 + Sqrt[1 - x])], {x, 0, n}, 
   Assumptions -> {n >= 1, n ∈ Integers}]

I get: mma-outputs-error which clearly asserts that the solution should be zero identically (and can be obtained by apply FullSimplify)– so, did Mathematica make a mistake?

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  • $\begingroup$ Please, try to be as descriptive as possible when asking questions. $\endgroup$
    – Sektor
    Commented Jul 16, 2015 at 12:45
  • $\begingroup$ What happens if you actually evaluate that beastie at nonnegative integers? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jul 16, 2015 at 15:48
  • $\begingroup$ clearly a b-g. SeriesCoefficient gives the correct result if you specify any particular n. $\endgroup$
    – george2079
    Commented Jul 16, 2015 at 16:00
  • $\begingroup$ This bug is present even in Math 7.0.1 on win7x64. $\endgroup$
    – innaiz
    Commented Mar 6, 2017 at 11:53

2 Answers 2

Reset to default
9
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Here is a workaround:

f[x_] = Log[1/2 (1 + Sqrt[1 - x])];

Generate a sequence from several of the coefficients

seq = Table[
   SeriesCoefficient[f[x], {x, 0, n}],
   {n, 8}];

Find the function that generates that sequence

coef[n_] = FindSequenceFunction[seq, n] //
  FullSimplify

-(Pochhammer[3/2, -1 + n]/ (4*n*Gamma[1 + n]))

For an alternative form for the coefficients

coef2[n_] = coef[n] // FunctionExpand

-(Gamma[1/2 + n]/(2*n^2*Sqrt[Pi]* Gamma[n]))

Verifying that these are the general coefficients for the original function

f[x] == Sum[coef[n]*x^n, {n, 1, Infinity}] == 
  Sum[coef2[n]*x^n, {n, 1, Infinity}] // Simplify

True

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  • $\begingroup$ Thanks a lot, but what's the reason mma give me the wrong outputs? $\endgroup$
    – van abel
    Commented Jul 17, 2015 at 0:19
  • $\begingroup$ Its algorithm failed; hence the need for a workaround. $\endgroup$
    – Bob Hanlon
    Commented Jul 17, 2015 at 0:43
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Interesting, doing nearly the same thing as @BobHanlon:

 clist = CoefficientList[
        Normal@Series[Log[1/2 (1 + Sqrt[1 - x])], {x, 0, 12}], x];
 FindSequenceFunction[clist, n]

yields a (Correct) DifferenceRoot expression:

DifferenceRoot[ Function[{[FormalY], [FormalN]}, {(-1 + 3 [FormalN] - 2 [FormalN]^2) [FormalY][[FormalN]] + 2 [FormalN]^2 [FormalY][1 + [FormalN]] == 0, [FormalY][1] == 0, [FormalY][2] == -(1/4)}]][n]

The difference here is clist includes the 0 first term. Not so useful but maybe it will give some insight to the cause of the error.

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1
  • $\begingroup$ Use FindSequenceFunction[Rest@clist, n] // FullSimplify and you will get the same result as mine. $\endgroup$
    – Bob Hanlon
    Commented Jul 16, 2015 at 17:49

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