Answer
The series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{ (2n+1)!}$, converges.
Work Step by Step
For the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{ (2n+1)!}$, the positive term $b_n=\frac{1}{ (2n+1)!}$
The terms are decreasing and positive.
Evaluate the limit:
$$\lim_{n\to \infty}b_n=\lim_{n\to \infty}\frac{1}{ (2n+1)!}=\frac{1}{\infty}=0$$
Thus, by the alternating series test, the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{ (2n+1)!}$, converges.
