A proof for Gauss' test for convergence

[raopeng]

Homework Statement

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 $\frac{c}{n^p}$.

Homework Equations

Basic convergence test(Cauchy and d'Alembert's tests)

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.

 

[mighty2000]

Hello! Thank you for your post. This is a very interesting problem indeed. Here are a few hints to get you started:

1. Recall the definition of absolute convergence. What does it mean for a sequence to converge absolutely?

2. Consider the sequence b_n = \frac{1}{n^p}. What can you say about its convergence?

3. Now, let's look at the sequence c_n = \frac{a_n}{a_{n+1}} - 1. What can you say about its convergence?

4. Can you use the fact that β converges absolutely to show that c_n converges to 0?

5. Finally, use the basic convergence test to show that the sequence a_n is of the same order as \frac{c}{n^p} at infinity.

I hope these hints help you get started on the problem. Good luck!