Step 1: Understanding the Topic
This question asks whether a given sequence converges or diverges. A sequence converges if its terms approach a single, unique finite limit as $n$ goes to infinity. A common way to test for divergence is to see if different parts (subsequences) of the sequence approach different limits.
Step 2: Key Approach - Subsequence Analysis
The term $(-1)^n$ causes the sequence to behave differently for even and odd values of $n$. We will analyze the limit of two subsequences: one for even terms ($n=2k$) and one for odd terms ($n=2k+1$). If these subsequences do not converge to the same limit, the original sequence diverges.
Step 3: Detailed Explanation
A. Analyze the subsequence for even $n$:
Let $n=2k$, where $k$ is a positive integer. For even $n$, $(-1)^n = 1$. The terms of this subsequence are:
\[
a_{2k} = 1 - (1) + \frac{1}{2k} = \frac{1}{2k}
\]
As $k \to \infty$, the limit of this subsequence is:
\[
\lim_{k\to\infty} a_{2k} = \lim_{k\to\infty} \frac{1}{2k} = 0
\]
B. Analyze the subsequence for odd $n$:
Let $n=2k+1$. For odd $n$, $(-1)^n = -1$. The terms of this subsequence are:
\[
a_{2k+1} = 1 - (-1) + \frac{1}{2k+1} = 2 + \frac{1}{2k+1}
\]
As $k \to \infty$, the limit of this subsequence is:
\[
\lim_{k\to\infty} a_{2k+1} = \lim_{k\to\infty} \left(2 + \frac{1}{2k+1}\right) = 2 + 0 = 2
\]
C. Compare the limits:
The subsequence of even terms converges to 0, while the subsequence of odd terms converges to 2. Since the sequence has subsequences that converge to different limits, the sequence as a whole cannot converge.
Step 4: Final Answer
The sequence has two different limit points (0 and 2), therefore it is divergent.
\[
\boxed{\text{The sequence is Divergent.}}
\]