# A proof for Gauss' test for convergence

1. Apr 22, 2013

### raopeng

1. The problem statement, all variables and given/known data
if $\frac{a_n}{a_{n+1}}=1+\frac{p}{n}+β_n$ and β converges absolutely, then at the infinity the sequence a is of the same order as $\frac{c}{n^p}$.

2. Relevant equations
Basic convergence test(Cauchy and d'Alembert's tests)

3. The attempt at a solution
I find this question quite interesting because it sunsequently proves Gauss' test for convergence without resorting to Raabe's test, but now has no idea where to begin. I try to write it as $n(\frac{a_n}{a_{n+1}}-1)=p$but this basically returns to the Raabe's test. The question prior to this one asks to prove that if $\frac{a_n}{a_{n+1}}=1+β_n$, then a has a limit. But I cannot find any ways of applying it to this question. Thanks in advance.