Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = e ^ { n / ( n + 6 ) }$
A) $e$
B) $\ln 3$
C) diverges
D) $e ^ { 3 }$
E) $\ln \left( \frac { 1 } { 3 } \right)$

Question

Quizplus · Accepted Answer

We can use the fact that $e^x$ grows faster than any polynomial function as $x \rightarrow \infty$. Thus, we can say that $$\lim_{n \rightarrow \infty} \frac{n}{n+6} = 1$$ Using L'Hopital's Rule or simple dividing both the numerator and denominator by $n$, we see that the limit is 1. Thus, we have $$\lim_{n \rightarrow \infty} e^{n/(n+6)} = \lim_{n \rightarrow \infty} e^{1-6/(n+6)} = e$$ Thus, the sequence converges to $e$.

Determine Whether the Sequence Converges or Diverges an=en/(n+6)a _ { n } = e ^ { n / ( n + 6 ) }an​=en/(n+6)

Determine Whether the Sequence Converges or Diverges $a _ { n } = e ^ { n / ( n + 6 ) }$