To determine the radius of convergence for a power series an, we must perform the ratio test limn→∞|an+1/an|.
So, with an=x4n/(4n)!, we have
limn→∞| ( x4(n+1)/(4(n+1))! ) / ( x4n/(4n)!) |
= limn→∞ | ( x4n+4/(4n+4)! ) * ((4n)!/x4n) |
= limn→∞ | x4/( (4n+4)(4n+3)(4n+2)(4n+1) ) |
As n tends to infinity, the denominator tends to infinity and the numerator stays fixed (we assume a fixed x for the test, and vary n). Hence, the limit is zero. Since the limit is zero, we say that the Radius of Convergence is ∞ and the interval of convergence is (-∞,∞).