Quiz 4
Quiz 4
Friday, April 27, 2001
NAME:
Student Number:_{}
No._{} Marks _{}
(25 Marks) Determine whether the series
å_{k = 1}^{¥}[((k!)^{2})/(2k)!] is convergent or
divergent.
Solution.
Let
a_{n} = [((k!)^{2})/(2k)!].
Then a_{n+1} = [(((k+1)!)^{2})/((2(k+1))!)] and



((k+1)!)^{2} (2(k+1))!

× 
(2k)! (k!)^{2}


 


((k+1)!)^{2} (k!)^{2}

× 
(2k)! (2(k+1))!


 


((k+1)k!)^{2} (k!)^{2}

× 
(2k)! (2k+2)(2k+1)(2k)!


 

 

 



It follows from the Ratio Test that
å_{k = 1}^{¥}[((k!)^{2})/(2k)!] is convergent.
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On 12 May 2001, 09:14.