Determine whether the sequence converges or diverges. if it converges, find the limit. (if an answer does not exist, enter dne.) an = (5^n + 1)/ 9^n

As , both terms will approach 0, so the sequence converges. To show this, we can invoke the monotone convergence theorem.

Consider the function , where . We have



and since for any such , and , we have that , which means the function is decreasing over its entire domain.

To recap: for all , so both and are bounded below. Then for all , which means and are decreasing.

Therefore by the monotone convergence theorem, both sequences converge to 0, and so must .

Exponential function