radius and convergence

Question

determine the interval and radius of convergence for the power series Σ n=1 to ∞ of (x^(4n))/(4n)!

Answer 1

Hi Kee,

 

To determine the radius of convergence for a power series a_n, we must perform the ratio test lim_n→∞|a_n+1/a_n|.

 

So, with a_n=x⁴ⁿ/(4n)!, we have

 

lim_n→∞| ( x⁴⁽ⁿ⁺¹⁾/(4(n+1))! ) / ( x⁴ⁿ/(4n)!) |

 

= lim_n→∞ | ( x⁴ⁿ⁺⁴/(4n+4)! ) * ((4n)!/x⁴ⁿ) |

 

= lim_n→∞ | 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 (-∞,∞).