Absolute Convergence Prove that if $\vert \frac{a_{n+1}}{a_n} \vert \leq \vert \frac{b_{n+1}}{b_n} \vert $ for $ n >>1$, and $\sum b_n$ is absolutely convergent, then $\sum a_n$ is absolutely convergent. 
My attempt at a solution:
By the Comparison Theorem, since $\vert \frac{b_{n+1}}{b_n} \vert $ converges by hypothesis, and since $\vert \frac{a_{n+1}}{a_n} \vert \leq \vert \frac{b_{n+1}}{b_n} \vert $, then $\sum a_n$ must also converge. 
Questions:
I know this is valid if $b_n$ is not equal to 1. However, how would I proceed to prove this in the case where it is equal to 1? Would I be able to make an estimate using the given information about $b_n$ being absolutely convergent?
Thanks in advance, I greatly appreciate suggestions/feedback/advice.
I am using the textbook Introduction to Analysis by Arthur Mattuck. 
 A: Here is a hint:
You have some $N$ such that For $n \ge N$, $| \frac{a_{n+1}}{a_n} | \le | \frac{b_{n+1}}{b_n} | $.
Then for $n > N$ (notice the strict inequality),  and we have $\frac{a_{n}}{a_N} = \frac{a_{N+1}}{a_{N}} \cdots \frac{a_{n}}{a_{n-1}}$, and similarly for the $b_n$.
So $|a_n| \le |a_N| |\frac{a_{N+1}}{a_{N}}| \cdots | \frac{a_{n}}{a_{n-1}} | \le |a_N| |\frac{b_{N+1}}{b_{N}}| \cdots | \frac{b_{n}}{b_{n-1}} | = |a_N| |\frac{b_{n}}{b_{N}}| = | \frac{a_N}{b_N} | |b_n|$.
Addendum: To see why $|\frac{b_{n+1}}{b_n}|$ does not necessarily converge, consider the following:
It is straightforward to see (because of the $\frac{1}{n^2}$ form) that $b_n = \begin{cases} \frac{4}{n^2}, & n \text{ even} \\ \frac{8}{n(n+1)^2}, & n \text{ odd} \end{cases}$ is a convergent series.
If $n$ is odd, we have $\frac{b_{n+1}}{b_n} = \frac{n}{2}$, and hence $\limsup_n \frac{b_{n+1}}{b_n} = \infty$. (And since $\liminf_n \frac{b_{n+1}}{b_n} = 0$, the ratio does not converge.)
A: @copper.hat Given $\sum b_n$ is absolutely convergent we must have at least $\lim_{n\to\infty} |\frac{b_{n+1}}{b_n}|\leq 1$. There are absolutely convergent series with $\lim_{n\to\infty} |\frac{b_{n+1}}{b_n}| = 1$ so we can't say more than that without more information. But, none the less, $|\frac{b_{n+1}}{b_n}|$ converges.
Apologies for this mistake and also for posting this as an answer. I had thought I was just posting a comment, but it appears as I'm new to Stack Exchange I'd missed the fact that I don't have the reputation to post comments. My arrival is heralded with various blunders!!
