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Half angle cosine sequence its limit at infinity!

Source: Indian Statistical Institute UGB 2023/2

May 14, 2023
trigonometrylimitscalculus

Problem Statement

Let a0=12a_0 = \frac{1}{2} and ana_n be defined inductively by an=1+an−12, n≥1.a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{, $n \ge 1$.}
[*] Show that for n=0,1,2,…,n = 0,1,2, \ldots, an=cos⁡(θn) for some 0<θn<π2, a_n = \cos(\theta_n) \text{ for some $0 < \theta_n < \frac{\pi}{2}$, } and determine θn\theta_n.
[*] Using (a) or otherwise, calculate lim⁡n→∞4n(1−an). \lim_{n \to \infty} 4^n (1 - a_n).
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