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 use the limit comparison test, comparing with the function $\frac{1}{8^n}.$
$$\lim_{n 	o \infty} \frac{\frac{5^n}{8^n+1}}{\frac{1}{8^n}} = \lim_{n 	o \infty} \frac{5^n}{8^n+1} \cdot \frac{8^n}{1}=\lim_{n 	o \infty} \frac{5^n}{8^n(1+\frac{1}{8^n})} = 0.$$ 
Since $\sum \frac{1}{8^n}$ is convergent, we have that $\sum a_n$ converges by limit comparison, and the limit is 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 $a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }$