Verified answer

This is equivalent to finding the interval of convergence of this series.

Since J₀(x) = 1 + ∑(n=1 to ∞) (-1)^n x^(2n) / [(n!)^2 * 2^(2n)], applying the Ratio Test yields:

r = lim(n→∞) |(-1)^(n+1) x^(2n+2) / [((n+1)!)^2 * 2^(2n+2)] / {(-1)^n x^(2n) / [(n!)^2 * 2^(2n)]}|

..= x^2 * lim(n→∞) 1 / [2^2 * (n+1)^2)]

..= 0.

Since r = 0 < 1, this series converges for all x.

Hence, the domain for J₀(x) is all real x.

I hope this helps!

All real numbers