Ferrus wrote: ↑Sat Jul 03, 2021 10:08 am
I will just graph a*sinh(1/2a) - 1, graphing that I get 0.23 as the minimum.
Actually this kind of intrigued me as to the best way to get this without blindly looking at a graph.
The Maclaurin series for sinh is:
SIGMA x^(2n+1)/(2n + 1)!
for the first term that is bluntly a(1/2a) - 1 = -1/2 not even close
for the second term you get:
a*(1/2a + 1/(8*3!)a^3) - 1= -a^2/2 + 1/48
which gives the quadratic solution of 0.204...
https://www.symbolab.com/solver/quadrat ... F48%20%3D0
Close but no cigar.
if you take the third term you get:
a*(1/2a + 1/(8*3!)a^3 + 1/(32*5!)a^5) - 1 = 1/2 + 1/48a^2 + 1/3840a^4 - 1 = -a^4/2 + a^2/48 + 1/3840 = 0.
https://keisan.casio.com/exec/system/1181809416
the only real positive root is 0.227, rounded up to the 2 d.p. is 0.23 which is the closest
So, in case anyone thinks graphing is cheating this is a hand calculated justification for the number.
