Determine whether the sequence defined by $a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges. If it converges, find its limit.
A) 1
B) $\frac { 5 } { 8 }$
C) 0
D) Diverges

Question

Quizplus · Accepted Answer

We can see that the denominator approaches infinity as n approaches infinity since 8^n gets larger much faster than 1. Thus, the limit of the sequence is 0. We can confirm this by applying the limit definition of convergence: 
$\lim_{n	o\infty}a_n=\lim_{n	o\infty}\frac{5^n}{8^n+1}=0$ because the numerator is growing, but the denominator is growing much faster. Thus, the answer is (C) 0.

Determine Whether the Sequence Defined By an=5n8n+1a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }an​=8n+15n​

Determine Whether the Sequence Defined By $a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }$