weird series: Z_3 = 1/2, Z_n= Z_{n+1}/[cos(pi/n)]. It's a bizarre suite decreasing to 0 without ever going.

[extrazlove]

hello

Here is a Z_n Suite



Z₃ = 1/2
Z_n= Z_n+1 / \cos (pi / n)


It's a bizarre suite decreasing to 0 without ever going.




I will define the n with which I work in my suite so as not to fall into absurdities like 1/0 which gives false calculations.
find that


n = pi / arccos (Z_n+1 / Z_n)


Note that Z_n #0 and Z_n+1/ Z_n #1
so the Z_n+1 limit is non-zero.
And since I have a decreasing sequence, Z_n+1 / Z_n < 1, so the series Z_n is convergent according to the d'Alembert criterion.

Is there an error in this reasoning?

Z_n is a positive term and is decreasing and minus 0 tends to 0 minus bound 0, which is impossible because Z_n #0?

 

[HallsofIvy]

So \(\displaystyle Z_3= 1/2\) and \(\displaystyle Z_n= \frac{Z_{n+1}}{cos(\pi/n)}\)? That second equation is oddly written. I would write it as \(\displaystyle z_{n+ 1}= z_n cos(pi/n)\).

You say "Z_n is a positive term and is decreasing and minus 0 tends to 0 minus bound 0, which is impossible because Z_n #0?"
\(\displaystyle Z_n\) is never 0 but that surely does not mean that the limit cannot be 0. The very simple sequence \(\displaystyle \{\frac{1}{n}\}\) has that same property: no term is ever 0 but the limit is 0.