The discussion revolves around finding sequences \( a_n \) and \( b_n \) such that the series \( \sum_{n=0}^{\infty} a_n \) and \( \sum_{n=0}^{\infty} b_n \) converge, while the series \( \sum_{n=0}^{\infty} \left( \sqrt{a_n} \cdot b_n \right) \) diverges. This involves exploring properties of convergent series and their products.
Some participants have expressed frustration in finding suitable sequences and have been encouraged to modify their approaches. There is an ongoing exploration of how to adjust the convergence rates of the series while considering the implications for their product.
Participants are required to demonstrate their efforts before receiving further assistance, which has led to discussions about acceptable forms of \( a_n \) and \( b_n \) and the need for creative modifications to achieve the desired outcome.
AfterSunShine said: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 so bad as a start. Can you modify the series in order to keep them converging, but doing so significantly slower?AfterSunShine said:basically I tried a_n = 1/n^2 & b_n = (-1)^n / n but failed