Proof About Series Convergence

[omert]

Homework Statement

let an be a positive series.
it is known that for every bn[tex]\rightarrow[/tex] [tex]\infty[/tex]
the sum from 1 to inf of an/bn is convergent
prove that the sum from 1 to inf of an is convergent

Homework Equations

The Attempt at a Solution

I thoght maybe to try to say. let [tex]\epsilon>0[/tex] and to say
that the sum of (an/bn)<epsilon
and then to choose bn in a certin way and by cauchy's cretertion to say
the it's convergent but I don't really know how

 

[mighty2000]

to show that the sum of an is convergent.

Thank you for your question. I would like to offer a proof for the convergence of the series an, given the convergence of the series an/bn.

First, we know that for every bn→∞, the series ∑an/bn is convergent. This means that for any ε>0, there exists an N such that for all n>N, we have ∑an/bn<ε.

Now, let's choose a specific bn, say bn=n^2. This is a common choice for bn in these types of problems. Then, we have ∑an/n^2<ε for all n>N.

Next, we can use the comparison test to show that ∑an is convergent. Since an/bn is convergent, we know that there exists an M such that for all n>M, we have an/bn<c for some constant c. This means that for all n>M, we have an<c/bn. Since bn=n^2, we have an<c/n^2 for all n>M.

Now, we can use the comparison test with the series ∑c/n^2, which is known to be convergent, to show that ∑an is also convergent. Therefore, we have shown that if an/bn is convergent, then an is also convergent.

I hope this helps. Please let me know if you have any further questions.