The finish was quite a jump for me. I guess I could go and try to stare at your parenthesis and figure it out myself, but mostly I feel somewhat abandoned at that step. I was excited when I found 1, 2, 4, 8… = −1 to be making sense, but that excitement doesn’t quite feel sufficient for me to want to decode the relationships between the terms in those two(?) patterns and all the relevant values
That’s fair. What I was trying to convey (in my notation, deliberately annoying to reduce a sense of familiarity) is limx→0−∑∞k=1kekxcos(kx)=−112 . Any ideas for how I could write that better?
I added some actual values for concreteness. Hopefully that helps.
Do we have the value of the sum as a function of x, before going to the limit as x goes to 0 ? If yes, it would help (bonus points if it can be proven in a few lines).
The finish was quite a jump for me. I guess I could go and try to stare at your parenthesis and figure it out myself, but mostly I feel somewhat abandoned at that step. I was excited when I found 1, 2, 4, 8… = −1 to be making sense, but that excitement doesn’t quite feel sufficient for me to want to decode the relationships between the terms in those two(?) patterns and all the relevant values
That’s fair. What I was trying to convey (in my notation, deliberately annoying to reduce a sense of familiarity) is limx→0−∑∞k=1kekxcos(kx)=−112 . Any ideas for how I could write that better?
I added some actual values for concreteness. Hopefully that helps.
Do we have the value of the sum as a function of x, before going to the limit as x goes to 0 ? If yes, it would help (bonus points if it can be proven in a few lines).
Mathematica yields 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. That probably simplifies though.