# SeriesCoefficients expansion contradicts FullSimplify

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

• Please, try to be as descriptive as possible when asking questions. – Sektor Jul 16 '15 at 12:45
• What happens if you actually evaluate that beastie at nonnegative integers? – J. M.'s torpor Jul 16 '15 at 15:48
• clearly a b-g. SeriesCoefficient gives the correct result if you specify any particular n. – george2079 Jul 16 '15 at 16:00
• This bug is present even in Math 7.0.1 on win7x64. – innaiz Mar 6 '17 at 11:53

## 2 Answers

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

• Thanks a lot, but what's the reason mma give me the wrong outputs? – van abel Jul 17 '15 at 0:19
• Its algorithm failed; hence the need for a workaround. – Bob Hanlon Jul 17 '15 at 0:43

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] == 0, [FormalY] == -(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.

• Use FindSequenceFunction[Rest@clist, n] // FullSimplify and you will get the same result as mine. – Bob Hanlon Jul 16 '15 at 17:49