If someone would be willing to write like a step by step procedure in dumb people terms for how to find them both I would be so grateful I'm so lost so far I get that you do the ratio or the root test and then you do something with the answer to that
damn, i once knew all of this and you've made me notice how utterly and completely gone it now is from my brain due to never being used
I've learned for math I basically have to write myself a procedure in like 6th grade language bc I forget it immediately, the only problem is making an accurate procedure is hard
yeah but I've spent too much to fail out now, and I'm teetering
Paul's Online Math Notes. Go and know joy. Seriously the only way I got through college.
Once you apply the root test/ratio test you treat the resulting limit as a limit of a sequence to determine convergence/divergence. So yeah, just evaluate the limit at the endpoint. It's even better if you have a bounded & monotonic sequence as the result of the ratio/root test, so you want to try your best to evaluate a limit that has those two things, if you can discover it with some nifty algebra.
what is the process for these things?
So yeah, just evaluate the limit at the endpoint
how do I know if it's a
bounded & monotonic sequence
A simple process for finding the radius of convergence:
1.) Observe the point where the series is != 0 and set the sequence of the series to some "a_n" .
2.) Apply the ratio/root test
3.) Simplify as much as possible. The goal here being to simplify the sequence so you have some clear idea of how it behaves at the end points. I'm not really sure how to word this differently, but like if during the process of evaluating, lets say, a ratio test: lim n-> infinity |3(x-2)(n+1)/3(x-2)n| you can see that the 3's cancel, and (x-2)^n+1 / (x-2)^n simplifies to just |x-2| (since division of exponential is the same as subtracting their powers, you can review this in algebra)
4.) Using our previous little example |x-2| < 1 converges and diverges when |x-2| > 1 (these are just the property of the ratio test) so the radius of convergence is R = 1.
To answer the part about how do you know if it's a bounded and monotonic sequence:
A bounded sequence is a sequence which has a definite lower bound and definite upper bound. And a monotonic sequence is one which is either non-decreasing or non-increasing so {1/n} is bounded and monotonic because {1/n} is bounded above by a_1 (a_1 = 1) and bounded below by 0 since the sequence never gets to 0 and thus never goes lower than 0. There is not a single element in the sequence which is greater than the previous one, the sequence is {1, 1/2, 1/3 ...} , so it is non-increasing and that makes it monotonic.
Wolfram Alpha might help as well.
https://www.wolframalpha.com/input/?i=series+convergence
