Example

Determine whether the following sequences converge or diverge:

$\{ s_{n}\} =\left\{ \left( \dfrac{3}{4}\right) ^{n}\right\}$

$\{ t_{n}\} =\left\{\left( \dfrac{4}{3}\right) ^{n}\right\}$

Solution (a) The sequence $\{ s_{n}\} =\left\{ \left(\dfrac{3}{4}\right) ^{n}\right\}$ converges to $0$ because $-1<\dfrac{3}{4}<1.$

(b) The sequence $\{ t_{n}\} =\left\{ \left( \dfrac{4}{3}\right) ^{n}\right\}$ diverges because $\dfrac{4}{3}>1$ .