MathDB

Let

n > 1

be a fixed integer. Define functions f_0(x) \equal{} 0, f_1(x) \equal{} 1 \minus{} \cos(x), and for

k > 0,

f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x). If F(x) \equal{} \sum^n_{r\equal{}1} f_r(x), prove that (a)

0 < F(x) < 1

for 0 < x < \frac{\pi}{n\plus{}1}, and (b)

F(x) > 1

for \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.

Problem Statement