# Find a_n & b_n such that

## 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.

None.

## The Attempt at a Solution

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

berkeman
Mentor

## 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.

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...

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
Mentor
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...

berkeman