Can you explain why this series converges? $$1 - \frac{1}{3 \cdot 2!} + \frac{1}{5 \cdot 3!} - \frac{1}{7 \cdot 4!}+\cdots$$
I am new to math over flow, and I do not know how to format the math, sorry! Also, what should this converge to?

$$1 - \frac{1}{3 \cdot 2!} + \frac{1}{5 \cdot 3!} - \frac{1}{7 \cdot 4!}+\cdots$$
A standard theorem says this converges if the corresponding series of absolute values converges:
$$1 + \frac{1}{3 \cdot 2!} + \frac{1}{5 \cdot 3!} + \frac{1}{7 \cdot 4!}+\cdots$$
The "comparison test" says this converges if the following converges:
$$1 + \frac 1 {2!} + \frac 1 {3!} + \frac 1 {4!}+\cdots$$
And that converges by a ratio test:
$$
\frac{1/(n+1)!}{1/n!} = \frac 1 {n+1} \to \text{ as } n\to\infty.
$$
