91Ó°ÊÓ

Show that each of the following sequences \(\left\\{a_{n}\right\\}\) is convergent, and find its limit: (a) \(a_{n}=\frac{n^{3}+2 n^{2}+3}{2 n^{3}+1}\); (b) \(a_{n}=\frac{n^{2}+2^{n}}{3^{n}+n^{3}}\); (c) \(a_{n}=\frac{n !+(-1)^{n}}{2^{n}+3 n !}\).

(a) Use the Binomial Theorem to prove that $$ n^{1} \leq 1+\sqrt{\frac{2}{n-1}}, \quad \text { for } n=2,3, \ldots $$ Hint: \(\quad(1+x)^{n} \geq \frac{n(n-1)}{2 !} x^{2}, \quad\) for \(n \geq 2, x \geq 0\). (b) Use the Squeeze Rule to deduce that $$ \lim _{n \rightarrow \infty} n^{\frac{1}{n}}=1 $$

By considering the product of the first \(n\) terms of the sequence \(\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\}\), prove that \(n !>\left(\frac{n+1}{e}\right)^{n}\), for \(n=1,2, \ldots .\)

Classify each of the following statements as TRUE or Note this general approach to FALSE, and justify your answers (that is, if a statement is TRUE, prove it; 'TRUE' and 'FALSE'. if a statement is FALSE, give a specific counter-example): (a) The terms of the sequence \(\left\\{2^{n}\right\\}\) are eventually greater than 1000 . (b) The terms of the sequence \(\left\\{(-1)^{n}\right\\}\) are eventually positive. (c) The terms of the sequence \(\left\\{\frac{1}{n}\right\\}\) are eventually less than \(0.025\). (d) The sequence \(\left\\{\frac{n^{4}}{4^{4}}\right\\}\) is eventually decreasing.

Use the inequality \(2^{n} \geq n^{2}\), for \(n \geq 4\), and the Squeeze Rule to prove that the sequence \(\left\\{n\left(\frac{1}{2}\right)^{n}\right\\}\) is null.

Prove that the following sequences are null: (a) \(\left\\{\frac{1}{n^{2}+n}\right\\}\) (b) \(\left\\{\frac{(-1)^{n}}{n !}\right\\}\); (c) \(\left\\{\frac{\sin n^{2}}{n^{2}+2^{n}}\right\\} .\)