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

$$$$$$$$$$$$$$Un=ln\left(cos\frac{1}{2^n}\right)$$$$showthatUn⩽0$$$$$$$$$$$$$$$$$$

Answered by deleteduser1 last updated on 08/Dec/23

$$0⩽cos\left(\frac{1}{2n}\right)⩽1=e^0\Rightarrow U_n⩽ln\left(e^0\right)=0$$

Answered by Mathspace last updated on 08/Dec/23

$$wehave0⩽\frac{1}{2^n}<\frac{\pi }{2}\mathrm{\forall }n⩾1\Rightarrow$$$$0<cos\left(\frac{1}{2^n}\right)⩽1\Rightarrow ln\left(cos\frac{1}{2^n}\right)⩽0\Rightarrow$$$$u_n⩽0$$

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