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!