30. {\\frac{e^{n}+\\pi^{n}}{e^{n}}}
30. {\\frac{e^{n}+\\pi^{n}}{e^{n}}}
Answer
Explanation:
Step1: Simplify the sequence term
$\frac{e^{n}+\pi^{n}}{e^{n}}=\frac{e^{n}}{e^{n}}+\frac{\pi^{n}}{e^{n}} = 1 + (\frac{\pi}{e})^{n}$
Step2: Find the limit as $n\rightarrow\infty$
We know that $\frac{\pi}{e}\approx\frac{3.14}{2.718}> 1$. As $n\rightarrow\infty$, $\lim_{n\rightarrow\infty}(\frac{\pi}{e})^{n}=\infty$. So $\lim_{n\rightarrow\infty}(1 + (\frac{\pi}{e})^{n})=\infty$.
Answer:
The sequence $\left{\frac{e^{n}+\pi^{n}}{e^{n}}\right}$ diverges.