Originally Posted by

**process91** I just got done proving Gauss' test, which is given in the book as:

If there is an $\displaystyle N\ge 1$, an $\displaystyle s>1$, and an $\displaystyle M>0$ such that

$\displaystyle \frac{a_{n+1}}{a_n}=1 - \frac{A}{n} + \frac{f(n)}{n^s}$

where $\displaystyle |f(n)|\le M$ for all n, then $\displaystyle \sum a_n$ converges if $\displaystyle A>1$ and diverges if $\displaystyle A \le 1$.

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

$\displaystyle \sum_{n=1} ^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot (2n)} \right)^k$

converges if $\displaystyle k>2$ and diverges if $\displaystyle k \le 2$ using Gauss' test.

OK, so much for the preamble. Here's my attempt:

$\displaystyle \frac{a_{n+1}}{a_n} = \left ( \frac {2n+1}{2n+2} \right ) ^ k$

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 $\displaystyle \left (\frac{n}{n+1} \right)^k \approx 1 - \frac{k}{n}$ for large n, but the book has not covered anything like that (it introduced the ~ symbol, but not $\displaystyle \approx$).