The problem asks us to analyze the sequence $a_n$ defined by $a_n = \frac{(-\pi)^n}{5^n}$. The most likely question is whether this sequence converges or diverges, and if it converges, what is its limit?

1. Problem Description

The problem asks us to analyze the sequence $a_n$ defined by $a_n = \frac{(-\pi)^n}{5^n}$. The most likely question is whether this sequence converges or diverges, and if it converges, what is its limit?

2. Solution Steps

We can rewrite the expression for $a_n$ as:

$a_n = \frac{(-\pi)^n}{5^n} = \left( \frac{-\pi}{5} \right)^n$

This is a geometric sequence with common ratio $r = -\frac{\pi}{5}$. A geometric sequence $a_n = r^n$ converges if and only if $|r| < 1$. In our case, $r = -\frac{\pi}{5}$, so we need to check if $|-\frac{\pi}{5}| < 1$.

$|-\frac{\pi}{5}| = \frac{\pi}{5}$

Since $\pi \approx 3.14159$, we have $\frac{\pi}{5} \approx \frac{3.14159}{5} \approx 0.6283$.

Since $0.6283 < 1$, we have $|\frac{-\pi}{5}| < 1$.

Thus, the sequence converges, and its limit is

0.

The general formula for the limit of a geometric sequence is:

$\lim_{n \to \infty} r^n = 0$ if $|r| < 1$.

3. Final Answer

The sequence converges to

0.