I just got done proving Gauss' test, which is given in the book as:
If there is an , an , and an such that
where for all n, then converges if and diverges if .
This is equivalent to the many other forms I have found on the web. The next question asks to use this test to prove that the series
converges if and diverges if using Gauss' test.
OK, so much for the preamble. Here's my attempt:
And that's it... I don't know how to put this into a form which corresponds in general to the form required for Gauss' test. I saw a similar problem online solved with the use of the fact that for large n, but the book has not covered anything like that (it introduced the ~ symbol, but not ).
Thank you very much for your reply. I'm still a little confused; either the book has not yet covered anything to explain the steps in both (2) and (3), or I missed it (this is Calculus Vol I. by Apostol). It did cover little-o notation in the context of Taylor Series, but that was only useful at the time because we were considering . I see that
, so I understand how the statement is true, but have no idea how to have come up with attempting that on my own. Is this a known result that one will apply? Where could I read more about this?
In step (3), I am completely at a loss for how to proceed. I don't know what to do with the little-o in the exponent, and I'm not sure how you got your result.
I think that the answer to this question requires external knowledge about little-o notation that I do not have yet. I don't want to take up so much of your time, if you would be so kind as to point me to another reference I would appreciate it.