Show a series converges for any x

Is there a good method for showing that a series converges, if I can't describe the series in general?

That is, if I know the pattern of the progression, but don't know how to express it for the nth term in terms of elementary functions, is there a surefire way to show it converges?

Take, for instance, a progression like:

1 - (1/2)x^2 + (1/2)(1/4)x^4 - (1/2)(1/4)(1/6)x^6 + ...

I know the pattern, but I can't figure out how to express it in terms of factorials and whatnot.

Any help is appreciated!

Hello, primasapere!

Quote:

---

---

We can express the denominators.

For example: .

So the general term is: . . for