Question

Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \] is convergent or divergent.

Hint:

If even and odd subsequences approach different limits, the sequence diverges.

Answer 1

Step 1: Examine $(-1)^n$.
If $n$ is even: \[ (-1)^n = 1. \] \[ a_n = 1 - 1 + \frac{1}{n} = \frac{1}{n}. \]
If $n$ is odd: \[ (-1)^n = -1. \] \[ a_n = 1 - (-1) + \frac{1}{n} = 2 + \frac{1}{n}. \]
Step 2: Find subsequence limits.
For even $n$: \[ a_{2k} \to 0. \]
For odd $n$: \[ a_{2k+1} \to 2. \]
Step 3: Conclusion.
Since two subsequences approach different limits (0 and 2), the sequence does not have a unique limit.
\[ \boxed{\text{The sequence is Divergent.}} \]