Calculus 8th Edition

(a) A convergent sequence is a sequence that has a finite limit (i.e. $\lim_{n \to \infty} a_n=L$). Examples: $$a_n = \frac{1}{n},\space\space b_n = \frac{n}{n+1}$$ (b) A divergent sequence is any sequence that doesn't converge. It can oscillate between two values or shoot off to $\pm\infty$. Examples: $$a_n=n,\space\space b_n=\sin(\pi n)$$
(a) \begin{aligned}\lim\limits_{n \to \infty}a_n = \lim\limits_{n \to \infty}\frac{1}{n} &= \frac{1}{\infty}\\&=0\end{aligned} Which is a finite number, so $a_n$ is a convergent sequence. \begin{aligned}\lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty}\frac{n}{n+1}&=\lim\limits_{n \to \infty}\frac{1+1/n}{1}\\&=\frac{1+1/\infty}{1}\\&=\frac{1+0}{1}\\&=1 \end{aligned} Which is a finite number, so $b_n$ is a convergent sequence. (b) \begin{aligned}\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}n&=\infty\end{aligned} Since $a_n$ doesn't have a finite limit (it shoots off to $+\infty$), it is a divergent sequence. $$\lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty}\sin(\pi n)$$ Since $\sin(\pi n)$ oscillates between $-1$ and $1$, it doesn't tend towards a single finite number, so $b_n$ is a divergent sequence.