Determine whether the nonincreasing sequence converges or diverges.
-$$a _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

Taking the ratio of the terms, we have:
$$\frac{a_{n+1}}{a_n}=\frac{8^{n+2}+7^{n+1}}{(n+1)\cdot8^{n+1}}\cdot\frac{n\cdot8^n}{8^{n+1}+7^n}$$
Rearranging the terms, we get:
$$\frac{a_{n+1}}{a_n}=\frac{8}{n+1}\cdot\frac{\left(1+\frac{7^n}{8^{n+1}}\right)}{\left(1+\frac{7^n}{8^{n+1}}\right)+\frac{1}{8}}$$
Since $\frac{7^n}{8^{n+1}}$ approaches zero as $n$ approaches infinity, we have:
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{8}{n+1}\cdot\frac{1}{1+\frac{1}{8}}=\lim_{n\to\infty}\frac{64}{9(n+1)}=0<1$$
Thus, by the ratio test, the series converges.

Determine Whether the Nonincreasing Sequence Converges or Diverges an=8n+1+7nn⋅8na _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }an​=n⋅8n8n+1+7n​

Determine Whether the Nonincreasing Sequence Converges or Diverges $a _ { n } = \frac { 8 ^ { n + 1 } + 7 ^ { n } } { n \cdot 8 ^ { n } }$