Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. $a _ { n } = \frac { 3 ^ { n } } { 4 ^ { n } }$
A) The sequence diverges.
B) The sequence converges to 2 .
C) The sequence converges to $- 1$.
D) The sequence converges to 1 .
E) The sequence converges to 0 .

Question

Quizplus · Accepted Answer

We can rewrite the nth term as:

$$a_n = \frac{3^n}{4^n} = \left(\frac{3}{4}ight)^n$$

Since $0<\frac{3}{4}<1$, we know that $\lim_{n	o\infty}\left(\frac{3}{4}ight)^n=0$ by the geometric series test. Therefore, the sequence converges to 0. Choice E is the correct answer.

Determine the Convergence or Divergence of the Sequence with the Given