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 ).