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April 7th, 2017, 07:29 PM   #1
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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?
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April 7th, 2017, 08:40 PM   #2
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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 non-recursive 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 non-recursive form for the odd sequence and to determine the radii of convergence.
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