isi-entrance 2019 Q6

isi-entrance · India · UGB Taylor series Limit evaluation using series expansion or exponential asymptotics

For all natural numbers $n$, let $$A_{n} = \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}} \quad (n \text{ many radicals})$$
(a) Show that for $n \geq 2$, $$A_{n} = 2\sin\frac{\pi}{2^{n+1}}$$
(b) Hence, or otherwise, evaluate the limit $$\lim_{n \rightarrow \infty} 2^{n} A_{n}$$

For all natural numbers $n$, let
$$A_{n} = \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}} \quad (n \text{ many radicals})$$

(a) Show that for $n \geq 2$,
$$A_{n} = 2\sin\frac{\pi}{2^{n+1}}$$

(b) Hence, or otherwise, evaluate the limit
$$\lim_{n \rightarrow \infty} 2^{n} A_{n}$$

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8