Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \sqrt [ n ] { 6 ^ { n } \cdot n }$$
A) 1
B) 6
C) 0
D) Diverges

Question

Quizplus · Accepted Answer

The limit of the sequence $\sqrt[n]{6^n \cdot n}$ as $n$ approaches infinity can be found by analyzing the expression inside the nth root. The dominant term here is $6^n$, as it grows much faster than $n$ as $n$ approaches infinity. Therefore, the limit of the sequence is essentially determined by the limit of $\sqrt[n]{6^n}$, which is 6. This is because, as $n$ increases, the impact of multiplying by $n$ inside the nth root becomes negligible, and the expression behaves similarly to $\sqrt[n]{6^n}$, which converges to 6.

Find the Limit of the Sequence If It Converges; Otherwise an=6n⋅nna _ { n } = \sqrt [ n ] { 6 ^ { n } \cdot n }an​=n6n⋅n​

Find the Limit of the Sequence If It Converges; Otherwise $a _ { n } = \sqrt [ n ] { 6 ^ { n } \cdot n }$