Determine the interval of convergence

I used the ratio test and got it down to

but I think I messed up somewhere or maybe I shouldn't use the ratio test and maybe try something different.

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- November 12th 2008, 11:54 PMkl.twillegerDetermine the Interval of Convergence
Determine the interval of convergence

I used the ratio test and got it down to

but I think I messed up somewhere or maybe I shouldn't use the ratio test and maybe try something different. - November 13th 2008, 10:12 AMo_O
Is this your series:

Using the ratio test:

So by the ratio test, your series absolutely converges and thus converges if

Simply solve. - November 13th 2008, 01:50 PMMathstud28
You don't have to use the Ratio test. Just note that

Which is a geometric series and converges iff the value raised to the n is less then one.

P.S. Dont forget to check the endpoints of your IOC

Also note that for every x that makes this seris converge it is euqal to - November 13th 2008, 01:58 PMJameson
What test is "needed" I think really depends on the level of rigorousness expected for the problem. Your observation about a_k is a good intuitive insight but without justification it's useless in the context of a proof. I'm sure you can easily show why what you said is true but like I said it all depends on the rigorousness required of a solution.

- November 13th 2008, 02:01 PMMathstud28
Yes, I understand what you are saying. A lot of the things I say similar to this are not meant to be taken as solutions, or even as what I would put down. They are more a way to rationalize why the answer is what it is. It also enables the student to form a more intuitive insight into mathematics as a whole. So yes you are correct, this should never be given as a solution especially in a formal situation like a quiz or test but I beleive that making these "intuitive insights" is the key to advanced mathematical success.