1. **Problem statement:** Test the convergence or divergence of the series \(\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n+1)!} \). 2. **Formula and rules:** For series with factorials and alternating signs, the Alternating Series Test and Ratio Test are useful. 3. **Step 1: Apply the Ratio Test:** Calculate \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) where \(a_n = \frac{(-1)^{n+1}}{(2n+1)!}\). \[ L = \lim_{n \to \infty} \frac{1/(2(n+1)+1)!}{1/(2n+1)!} = \lim_{n \to \infty} \frac{(2n+1)!}{(2n+3)!} = \lim_{n \to \infty} \frac{1}{(2n+2)(2n+3)} = 0 \] 4. Since \(L=0 < 1\), the Ratio Test confirms the series converges absolutely. 5. **Interpretation:** The factorial in the denominator grows very fast, making terms approach zero quickly, ensuring convergence. **Final answer:** The series \(\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n+1)!} \) converges absolutely.