8 Multiple Choice 1 point
Determine the convergence or divergence of the series using the Ratio Test:

Ask by Mccarthy Frank. in the United States

Mar 27,2025

Question

8 Multiple Choice 1 point Determine the convergence or divergence of the series using the Ratio Test: $$ \sum_{n=1}^{\infty} \frac{\pi_{n}^{n}}{6^{n}} $$

UpStudy AI · Accepted Answer

Let

$$
a_{n}=\frac{\pi^{n}}{6^{n}}.
$$

Using the Ratio Test, we consider

$$
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|.
$$

First, express $$ a_{n+1} $$:

$$
a_{n+1} = \frac{\pi^{n+1}}{6^{n+1}}.
$$

Now, form the ratio:

$$
\left|\frac{a_{n+1}}{a_n}\right| = \frac{\pi^{n+1}}{6^{n+1}} \cdot \frac{6^{n}}{\pi^{n}} = \frac{\pi^{n+1}}{\pi^{n}} \cdot \frac{6^{n}}{6^{n+1}} = \frac{\pi}{6}.
$$

Since the expression $$\frac{\pi}{6}$$ does not depend on $$ n $$, the limit is

$$
L = \frac{\pi}{6}.
$$

We know that

$$
\frac{\pi}{6} \approx \frac{3.14}{6} \approx 0.5236,
$$

which is less than 1. By the Ratio Test, if $$ L < 1 $$, the series

$$
\sum_{n=1}^{\infty} \frac{\pi^{n}}{6^{n}}
$$

converges absolutely.
The series $$ \sum_{n=1}^{\infty} \frac{\pi^{n}}{6^{n}} $$ converges absolutely.

8 Multiple Choice 1 point
Determine the convergence or divergence of the series using the Ratio Test:

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