Question

Let xn = ( 2 + (−1)^n/ n+2 ). Using the algebra of limits and standard results on limits to establish/justify if the sequence {xn} n=1 to infinity converges or diverges; find the limit in case the sequence is convergent.

Answer 1

$\lim \limits_{n \to \infty} \frac{2+(-1)^{n}}{n+2} = \lim \limits_{n \to \infty} \frac{\frac{2}{n} + \frac{(-1)^n}{n}}{\frac{n}{n}+ \frac{2}{n}}$

But

$\lim \limits_{n \to \infty} \frac{2}{n} = 0\\ \lim \limits_{n \to \infty} \frac{(-1)^n}{n} = 0$

Also

$\lim \limits_{n \to \infty} \frac{n}{n} = \lim \limits_{n \to \infty} 1 = 1$

$\lim \limits_{n \to \infty} \frac{2+(-1)^{n}}{n+2} = \frac{0+0}{1+0} = 0$

The sequence is convergent and the convergent point is zero.