Let C(theta) = sum_{n=0}^{infty} frac{cos(nth | vedclass

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(D) Given $C(	heta) = \sum_{n=0}^{\infty} \frac{\cos(n	heta)}{n!}$.$C(0) = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \dot

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$C^{\prime}(	heta) 
eq 0$ for all $	heta \in \mathbb{R}$

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(D) Given $C(	heta) = \sum_{n=0}^{\infty} \frac{\cos(n	heta)}{n!}$.$C(0) = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots = e$.$C(\pi) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} = 1 - \frac{1}{1!} + \frac{1}{2!} - \dots = e^{-1}$.$(A)$ $C(0) \cdot C(\pi) = e \cdot e^{-1} = 1$ (True).$(B)$ $C(0) + C(\pi) = e + \frac{1}{e} \approx 2.718 + 0.368 = 3.086 > 2$ (True).$(C)$ $C(	heta) = 	ext{Re}\left(\sum_{n=0}^{\infty} \frac{e^{in	heta}}{n!}ight) = 	ext{Re}(e^{e^{i	heta}}) = 	ext{Re}(e^{\cos 	heta + i \sin 	heta}) = e^{\cos 	heta} \cos(\sin 	heta)$. Since $e^{\cos 	heta} > 0$ and $\cos(\sin 	heta) > 0$ (as $|\sin 	heta| \le 1  0$ (True).$(D)$ $C^{\prime}(	heta) = \sum_{n=1}^{\infty} -\frac{n \sin(n	heta)}{n!} = -\sum_{n=1}^{\infty} \frac{\sin(n	heta)}{(n-1)!}$. At $	heta = 0$,$C^{\prime}(0) = 0$. Thus,the statement $C^{\prime}(	heta) 
eq 0$ for all $	heta \in \mathbb{R}$ is False.

Let $C(\theta) = \sum_{n=0}^{\infty} \frac{\cos(n\theta)}{n!}$. Which of the following statements is $FALSE$?

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