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:

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$\displaystyle 1 - \frac{1}{2}x^2 + \frac{1}{2\cdot4}x^4 - \frac{1}{2\cdot4\cdot6}x^6 + \frac{1}{2\cdot4\cdot6\cdot8}x^8 -\hdots$

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We can express the denominators.

For example: .$\displaystyle 2\cdot4\cdot6\cdot8 \:=\:(2\cdot1)(2\cdot2)(2\cdot3)(2\cdot4) \:=\:2^4(1\cdot2\cdot3\cdot4) \:=\:2^4\cdot4!$

So the general term is: .$\displaystyle (-1)^n\frac{x^n}{2^n\cdot n!}$ . for $\displaystyle n = 0,1,2,3,\hdots$