Let C - n -0-cos n - n -Which of the following statements is FALSE- A- C -0 for all - RB- C -0 for all - RC- C 0- C -2D- C 0 - C -1

C(θ)=∑_n=0^∞cos (nθ)n!=(1+cosθ/1!+cos 2θ/2!+cos 3θ/3!+...+cos nθ/n!) C(0)=1+1/1!+1/2!+1/3!+..... upto ∞ term =e C(π)=1 1/1!+1/2! 1/3!+..... upto ∞ term ...

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Standard XII

Mathematics

Derivative of Standard Functions

Let C θ=∑ n =...

Question

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

C (0) . C (π) = 1

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C (0) + C (π) > 2

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C (θ) > 0

for all

θ \in R

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C (θ) \neq 0

for all

θ \in R

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Solution

The correct option is D $C (θ) \neq 0$ for all $θ \in R$
$C (θ) = \infty \sum n = 0 \frac{cos (n θ)}{n!} = (1 + \frac{cos θ}{1!} + \frac{cos 2 θ}{2!} + \frac{cos 3 θ}{3!} + . . . + \frac{cos n θ}{n!})$
$C (0) = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + . . . . . upto \infty term = e$
$C (π) = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + . . . . . upto \infty term = e^{- 1}$
So, $C (0) . C (π) = 1$ and
$C (0) + C (π) = e + \frac{1}{e} > 2$
$C^{'} (θ) = (- \frac{sin θ}{1!} - 2 \frac{sin 2 θ}{2!} - 3 \frac{sin 3 θ}{3!} + . . . + n \frac{sin n θ}{n!})$
Therefore at $θ = 0 \Rightarrow C^{'} (θ) = 0$

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

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