How do I find the interval of convergance for this?

I did the ratio test and get .25<x<.25

The book has .5<x<.5

I will do the endpoint test after I see why I'm not getting one half as my interval!Attachment 28828

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- July 14th 2013, 12:52 PMminneola24Finding interval of convergance
How do I find the interval of convergance for this?

I did the ratio test and get .25<x<.25

The book has .5<x<.5

I will do the endpoint test after I see why I'm not getting one half as my interval!Attachment 28828 - July 14th 2013, 01:11 PMemakarovRe: Finding interval of convergance
- July 14th 2013, 01:29 PMminneola24Re: Finding interval of convergance
Ok.

How exactly do I do that, and why? - July 14th 2013, 01:45 PMPlatoRe: Finding interval of convergance
I answered that here.

Use the root on - July 14th 2013, 01:46 PMSorobanRe: Finding interval of convergance
Hello, minneola24!

Quote:

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Then: .

Therefore: .

- July 14th 2013, 03:38 PMemakarovRe: Finding interval of convergance
I intended to use facts specifically about the radius of convergence of a

*power*series instead of tests that apply to any series.

(set k = n + 1, i.e., n = k - 1)

(set n = 2k, i.e., k = n/2)

where

Now, strictly speaking, we cannot use the formula

because some are equal to 0 (but see Soroban's answer on how to use ratio test for arbitrary series), but we can use

Since as , , so r = 1/2. If we did not convert the series to the form and took the nth root of , we would get r = 1/4.