Let C (θ) = displaystyle ∑n=0∞ ( cos (nθ)/n !) Which of the following statements is FALSE ?

Question

Let C (θ) = displaystyle ∑n=0∞ ( cos (nθ)/n !) Which of the following statements is FALSE ? (A) C(0)⋅ C(π)=1 (B) C(0)+C(π)> 2 (C) C(θ)> 0 for all θ ∈ R (D) C' (θ) ≠0 for all θ ∈ R

Tardigrade Admin · Accepted Answer

Given, C (θ)= displaystyle∑n=0∞ ( cos(nθ)/n!) C(0)= displaystyle∑n=0∞ (1/n!)=1+(1/1!)+(1/2!)+(1/3!)+ ldots=e C(π)= displaystyle∑n=0∞ (-1)n (1/n!) [∵ cos nπ=(-1)n] C(π)=1-(1/1!)+(1/2!)-(1/3!)+ ldots=e-1 (A)C(0).C(π)=e.e-1=1 True (B)C(0)+C(π)=e+(1/e)> 2 True (C)C(θ)> 0 ∀ θ ∈ R True (D)C'(θ)= displaystyle∑n=0∞- (n sin (nθ)/n!) ∴ C'(θ)=0 ⇒ θ=0 False

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

$C (0) \cdot C (π) = 1$

$C (0) + C (π) > 2$

$C (θ) > 0$ for all $θ \in R$

$C^{'} (θ) \neq = 0$ for all $θ \in R$

Q. Let C(θ)=n=0∑∞​n!cos(nθ)​ Which of the following statements is FALSE ?

C(0)⋅C(π)=1

C(0)+C(π)>2

C(θ)>0 for all θ∈R

C′(θ)=0 for all θ∈R

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

$C (0) \cdot C (π) = 1$

$C (0) + C (π) > 2$

$C (θ) > 0$ for all $θ \in R$

$C^{'} (θ) \neq = 0$ for all $θ \in R$