1. The problem asks us to show that the sequence $1, -1, 1, -1, 1, -1, \ldots$ defined by the general term $a_n = (-1)^{n+1}$ diverges. 2. Recall that a sequence converges if it approaches a single finite limit as $n \to \infty$. 3. The given sequence is $a_n = (-1)^{n+1}$, which alternates between $1$ and $-1$: $$a_1 = (-1)^2 = 1, \quad a_2 = (-1)^3 = -1, \quad a_3 = 1, \quad a_4 = -1, \ldots$$ 4. Since the terms keep switching between $1$ and $-1$, the sequence does not approach any single number. 5. Therefore, the sequence does not have a limit and hence diverges. Final answer: The sequence $a_n = (-1)^{n+1}$ diverges because it oscillates between $1$ and $-1$ and does not approach any finite limit.