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April 7th, 2017, 06:29 PM  #1 
Member Joined: Feb 2014 Posts: 91 Thanks: 1  Power Series
question 12 I need to find two linearly independent power series and radius of convergence. This test is coming very soon. The class is accelerating due to missed school days and I need to be really ready. Am I on the right track with this problem? 
April 7th, 2017, 07:40 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,582 Thanks: 1427 
Solving this you get two sequences corresponding to the even and odd terms of $a_{n+2}=\dfrac{4}{(n+2)(n+1)}a_n$ The even sequence depends on $a_0$, the odd on $a_1$ You can compute nonrecursive formulas for these two sequences to determine the radius of convergence. the even sequence for example is $a_{k}=\dfrac{4^k}{(2 k)!}a_0,~k=0,1,2,\dots$ (I've reindexed it) I leave you to compute the nonrecursive form for the odd sequence and to determine the radii of convergence. 

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