Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }$$
A) $\ln 4$
B) 0
C) 1
D) Diverges

Question

Quizplus · Accepted Answer

Taking the natural logarithm of $a_n$ gives:
$$\ln a_n = \frac{4}{n} \ln \left(\frac{4}{n}ight) = \frac{4}{n} \left(\ln 4 - \ln n ight) = \frac{4 \ln 4}{n} - 4 \frac{\ln n}{n}$$
As $n$ approaches infinity, both terms approach zero by the Squeeze Theorem. Therefore, $\ln a_n$ approaches $\frac{4 \ln 4}{n}$, and $a_n$ approaches $\boxed{	extbf{(C)}\ 1}$ by taking the exponential of both sides.

Find the Limit of the Sequence If It Converges; Otherwise an=(4n)4/na _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }an​=(n4​)4/n

Find the Limit of the Sequence If It Converges; Otherwise $a _ { n } = \left( \frac { 4 } { n } \right) ^ { 4 / n }$