How Can We Modify Convergent Series to Make Their Product Diverge?

[AfterSunShine]

Homework Statement

Find a_n & b_n such that \sum_{n=0}^{\infty}a_n & \sum_{n=0}^{\infty}b_n are convergent series, but \displaystyle \sum_{n=0}^{\infty} \left( \sqrt{a_n} \cdot b_n \right) diverges.

Homework Equations

None.

The Attempt at a Solution

Try too hard for this but still cannot find such a_n & b_n.

 

[berkeman]

Homework Statement

Find a_n & b_n such that \sum_{n=0}^{\infty}a_n & \sum_{n=0}^{\infty}b_n are convergent series, but \displaystyle \sum_{n=0}^{\infty} \left( \sqrt{a_n} \cdot b_n \right) diverges.

Homework Equations

None.

The Attempt at a Solution

Try too hard for this but still cannot find such a_n & b_n.

That is not an acceptable post. You must show your efforts before we can offer tutorial help. Show us what you have tried so far please...

 

[AfterSunShine]

There is nothing to show here
basically I tried a_n = 1/n^2 & b_n = (-1)^n / n but failed
tried a_n = 1/n^2 & b_n = arctan (1/n) but failed
and so on...

 

[mfb]

basically I tried a_n = 1/n^2 & b_n = (-1)^n / n but failed

That is not so bad as a start. Can you modify the series in order to keep them converging, but doing so significantly slower?
And then you'll need some trick to get the product diverging - something that changes the sign flip thing...