Question

Suppose f : (−1, 1) → $\R$ is an infinitely differentiable function such that the series $\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}$ converges to f(x) for each x ∈ (−1, 1), where,
$a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta$
for j ≥ 0. Then

• A. f(x) = 0 for all x ∈ (−1, 1)
• B. f is a non-constant even function on (−1, 1)
• C. f is a non-constant odd function on (−1, 1)
• D. f is NEITHER an odd function NOR an even function on (−1, 1)

Answer 1

Answer: (B)

f is a non-constant even function on (−1, 1)

Answer 2

The correct option is (B) : f is a non-constant even function on (−1, 1).