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DEMJ
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Homework Statement
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
Homework Equations
This is true or false.
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?