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**primasapere** the function y is defined as a the expansion of a series (which i know to be the bessel function, though i only know the expansion terms):

y = 1 - (2^-2)*x^2 + (2^-2)*(4^-2)*x^4 - (2^-2)*(4^-2)*(6^-2)*x^6 + ...

thus i believe the power series can be defined as:

y = sum((-1/4)^n*(n!)^2*x^(2*n), n = 0 -> inf)

my goal is to show this series converges for any value of x.

i can't seem to use the radius of convergence approach, because that requires taking the limit of the ratio of successive terms, which seems to be zero, and by my book's definition of this radius, abs(x) < R converges, abs(x) > R diverges, so the series would certainly diverge.

i'm not the most familiar with showing convergence in a series, so any help is appreciated.