Un-ln-cos-1-2-n-show-that-Un-0-

Question Number 201533 by Rodier97 last updated on 08/Dec/23

U n = l n (c o s \frac{1}{2^{n}})

s h o w t h a t U n ⩽ 0

Answered by AST last updated on 08/Dec/23

0 ⩽ c o s (\frac{1}{2 n}) ⩽ 1 = e^{0} \Rightarrow U_{n} ⩽ l n (e^{0}) = 0

Answered by Mathspace last updated on 08/Dec/23

w e h a v e 0 ⩽ \frac{1}{2^{n}} < \frac{π}{2} \forall n ⩾ 1 \Rightarrow

0 < c o s (\frac{1}{2^{n}}) ⩽ 1 \Rightarrow l n (c o s \frac{1}{2^{n}}) ⩽ 0 \Rightarrow

u_{n} ⩽ 0