Question

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

• A. $C\left(0\right)\cdot C\left(\pi\right)=1$
• B. $C\left(0\right)+C\left(\pi\right)>\,2$
• C. $C\left(\theta\right)>\,0 $ for all $\theta \in R$
• D. $C' \left(\theta\right) \ne0$ for all $\theta \in R$

Answer 1

Answer: (D)

Given, $C \left(\theta\right)=\displaystyle\sum_{n=0}^{\infty} \frac{\cos\left(n\theta\right)}{n!}$
$C\left(0\right)=\displaystyle\sum_{n=0}^{\infty} \frac{1}{n!}=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots=e$
$C\left(\pi\right)=\displaystyle\sum_{n=0}^{\infty} \left(-1\right)^{n} \frac{1}{n!} \left[\because \cos\,n\pi=\left(-1\right)^{n}\right]$
$C\left(\pi\right)=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\ldots=e^{-1}$
$\left(A\right)C\left(0\right).C\left(\pi\right)=e.e^{-1}=1 $ True
$\left(B\right)C\left(0\right)+C\left(\pi\right)=e+\frac{1}{e}>\,2$ True
$\left(C\right)C\left(\theta\right)>\,0 \forall \,\theta \,\in\,R $ True
$\left(D\right)C'\left(\theta\right)=\displaystyle\sum_{n=0}^{\infty}- \frac{n\,\sin\,\left(n\theta\right)}{n!}$
$\therefore C'\left(\theta\right)=0$
$\Rightarrow \theta=0$ False