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Calculus  II                                           Quiz #9                     May 9, 2005

Name_________________            R.  Hammack                  Score ______
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(1)     Find the interval of convergence of the power series Underoverscript[∑ , n = 1, arg3]4^n/n^(1/2)x^n.

Using the ratio test for absolute convergence, we get
ρ=Underscript[lim , n∞](| u_ (n + 1) |)/(| u_n |)=Underscript[lim , n∞](| 4^(n + 1)/(n + 1)^(1/2) x^(n + 1) |)/(| 4^n/n^(1/2) x^n |)=Underscript[lim , n∞](| x |^(n + 1))/(| x |^n)n^(1/2)/(n + 1)^(1/2)4^(n + 1)/4^n=Underscript[lim , n∞]4| x|n/(n + 1)^(1/2)=4|x|

For convergence, we need ρ=4|x|<1, so |x| < 1/4,  or -1/4 < x < 1/4.

Check endpoint x = 1/4
Underoverscript[∑ , n = 1, arg3]4^n/n^(1/2)x^n=Underoverscript[∑ , n = 1, arg3]4^n/n^(1/2)(1/4)^n=Underoverscript[∑ , n = 1, arg3]1/n^(1/2) (divergent p-series)

Check endpoint x = -1/4
Underoverscript[∑ , n = 1, arg3]4^n/n^(1/2)x^n=Underoverscript[∑ , n = 1, arg3]4^n/n^(1/2)(-1/4)^n=Underoverscript[∑ , n = 1, arg3](-1)^n/n^(1/2) (convergent alternating p-series)

Conclusion: The interval of convergence is [-1/4, 1/4)