I didn’t make any claim about limits. If you’re looking for rigor, you’re in the wrong place, as I tried to make clear in the introduction.
But (A) is true without any unconventional weirdness: ∑∞k=1kekxcos(kx)=e(1+i)x(e2ix−4e(1+i)x+e2x+e(2+2i)x+1)2(−ex+eix)2(−1+e(1+i)x)2 (from Mathematica), and limx→0− of that is −112.
I didn’t make any claim about limits. If you’re looking for rigor, you’re in the wrong place, as I tried to make clear in the introduction.
But (A) is true without any unconventional weirdness: ∑∞k=1kekxcos(kx)=e(1+i)x(e2ix−4e(1+i)x+e2x+e(2+2i)x+1)2(−ex+eix)2(−1+e(1+i)x)2 (from Mathematica), and limx→0− of that is −112.