(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [tex]a_k[/tex] is decreasing and it's limit is 0 as [tex]k \to \infty[/tex] and [tex]\sum_{k+1}^{\infty} b_k[/tex] converges conditionally, then [tex]\sum_{k=1}^{\infty} a_k b_k[/tex] converges

2. Relevant equations

This is true or false.

3. The attempt at a solution

I think it is false because if we let [tex]a_k = \frac{1}{\sqrt{k}}, b_k= \frac{(-1)^k}{\sqrt{k}}[/tex] we satisfy our initial conditions but [tex]a_k \cdot b_k = \frac{1}{k}[/tex] so [tex]\sum_{k=1}^{\infty} a_k b_k[/tex] diverges.

Is this correct?

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# Conv. Conditionally Proff

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