Question

Problem 1 Find the limits of the following sequences (i) an- {V2, V2V2, \V2V/2V2, (iiu )

I, II, and III please!!!!

Answer 1

(i)

$a_n = \sqrt{2\sqrt{2\sqrt{2,...}}}$

For large n,

$\Rightarrow a_n = \sqrt{2 a_n}$

$\Rightarrow a_n^2 = 2 a_n$

$\Rightarrow a_n = 2$ if $a_n \ne 0$

Thus, the limit of the sequence is 2.

(ii)

$b_n = \sqrt[n]{3^{2+3n}} = \left ( 3^{2+3n} \right )^{1/n} = 3^{2/n + 3}$

$\lim_{n \to \infty }b_n = \lim_{n \to \infty }3^{2/n + 3} = 3^3 = 27$

(iii)

$\lim_{n \to \infty }c_n = \lim_{n \to \infty } \frac{(2n-1)^2}{n^2 + n + 5} = \lim_{n \to \infty } \frac{n^2(2-1/n)^2}{n^2(1 + 1/n + 5/n^2)}$

$= \lim_{n \to \infty } \frac{(2-1/n)^2}{(1 + 1/n + 5/n^2)} = 4$