Determine Whether \({\{r^{n}\}}\) Converges or Diverges

Determine whether the following sequences converge or diverge:

1. \(\{ s_{n}\} =\left\{ \left( \dfrac{3}{4}\right) ^{n}\right\}\)
2. \(\{ t_{n}\} =\left\{\left( \dfrac{4}{3}\right) ^{n}\right\} \)

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 \(\dfrac{4}{3}>1\).