Proving Convergence of Positive Series Using Ratio Test: A Guide

[Jamin2112]

Homework Statement

I'm studying for my final next week (In other words, I'll be asking a lot of questions on here).

One practice problem from the book reads

Prove that, if ∑a_n and ∑b_n are series of positive terms with ∑b_n convergent and a_n/b_n --> 0, then ∑a_n is convergent.

Homework Equations

Whatever

The Attempt at a Solution

So, if a_n/b_n --> 0, then either

b_n ---> ∞ or -∞ AND a_n ---> some finite number;

or b_n ----> some finite number AND a_n ---> 0.

The second option is true given that ∑b_n is convergent.


? I don't. Help me with a good way of explaining this.

 

[someperson05]

Take the following with a grain of salt, and perhaps ignore it as I don't have much experience with formalized dealings with series. But I have an intuitive suggestion that may lead to the proof. (Or it could be just down right wrong.)

If $$\Sigma b_{n}$$ converges, I think that implies $$b_{n}$$ goes to zero.
(The contrapositive of the theorem that if a sequence does not go to zero, then the series diverges.)
Now if $$b_{n}$$ goes to zero, then this has some implications for what $$\frac{a_{n}}{b_{n}}$$ actually means. Namely it would seem that if $$b_{n}$$ is going to zero, then that would cause the fraction to go to infinity. Since it doesn't, this means $$a_{n}$$ is forced to behave in a very particular way.

I think there lies the proof.

 

[ystael]

This is simple -- don't think too hard. The condition $$a_n/b_n \to 0$$ is actually much stronger than you need. You know $$\textstyle\sum b_n$$ converges, and $$a_n/b_n \to 0$$ is a relatively strong way of saying that "$$a_n$$ is eventually smaller than $$b_n$$". This should remind you of a particular test for series convergence which you can use to prove that $$\textstyle\sum a_n$$ converges.