#4972 powerseries of acos has every odd coefficient sign-flipped, and of asin returns an unusable series

[David Scherfgen]

powerseries(acos(x),x,0) gives %pi/2 + x + x^3/6 + ... where the series is %pi/2 - x - x^3/6 - ...

Maxima 5.49post, SBCL 2.6.7.

(%i1) display2d:false$
(%i2) powerseries(acos(x),x,0);
(%o2) (2*'sum((genfact(2*i1-1,i1,2)*x^(2*i1+1))/((2*i1+1)*genfact(2*i1,i1,2)),i1,0,inf)
       +%pi)/2
(%i3) taylor(acos(x),x,0,5);
(%o3) %pi/2-x-x^3/6-(3*x^5)/40
(%i4) powerseries(asin(x),x,0);
(%o4) 'sum((genfact(2*i1-1,i1,2)*x^(2*i1+1))/((2*i1+1)*genfact(2*i1,i1,2)),i1,0,inf)
(%i5) genfact(1,1,2);
genfact: generalized factorial not defined for given arguments.

Expected: (%o2) should be %pi/2 - 'sum(...), matching (%o3). Every summand in (%o2) is positive, so the series it returns is %pi/2 + asin(x) rather than %pi/2 - asin(x).

Separately, the series in (%o2) and (%o4) cannot be evaluated: the term at i1 = 1 requires genfact(2*1-1,1,2), that is genfact(1,1,2), which Maxima's own genfact rejects as shown in (%i5). So powerseries(asin(x),x,0) returns a formally correct but inert expression, and powerseries(acos(x),x,0) returns one that is both inert and sign-wrong.