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.
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
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.
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.