Let C(\theta) = \sum ^{\infty} _{n= 0}{\cos(n\theta )}/n! Which of the following statements is FALSE ?

Correct Answer is : C′(θ)≠0 for all θ∊R

C(\theta) > 0 for all \theta ∊ R

C'(\theta) \neq0 for all \theta∊ R

KVPY SX SB Exam 2015 Question Paper

Question : 84 of 120

Marks: +1, -0

Let

C(θ) = ∑ ↖{∞} ↙{n= 0}{\cos(nθ )}/n!

Which of the following statements is FALSE ?

Solution:

C(θ) = ∑ ↖{∞} ↙{n= 0}{\cos(nθ )}/n!

=\lim↙{n→∞}(1+{\cosθ}/1!+{\cos2θ}/2!+{\cos3θ}/3!

+...+{\cosnθ}/n!)

C(0) =1+1/1!+1/2!+1/3!+...

up to ∞ term

C(π) =1-1/1!+1/2!-1/3!+...

up to ∞ term
Clearly

C(0) . C(π) = 1

C(0) . C(π) = e+1/e>2

C'(θ) = -{\sinθ}/1!- {2\sin2θ}/2!- {3\sin3θ}/3!+...

up to ∞ term
And that value is equal to zero at θ = 0

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