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[solved] Series problem

bkarpuz

New member
Jan 27, 2012
11
Dear MHB members,

here is my problem.
Suppose that $\{a_{n}\}$ and $\{b_{n}\}$ are sequences such that $a_{n}>b_{n}\geq0$ for all $n\in\mathbb{N}$.
Further, suppose that $\sum_{k=1}^{\infty}\frac{1}{a_{k}}=\infty$ and $\sum_{k=1}^{\infty}b_{k}<\infty$.
Prove or disprove $\prod_{k=1}^{\infty}\big(1-\frac{b_{k}}{a_{k}}\big)>0$.

Thank you.
bkarpuz

---------- Post added at 10:14 AM ---------- Previous post was at 10:00 AM ----------

Dear MHB members,

here is my problem.
Suppose that $\{a_{n}\}$ and $\{b_{n}\}$ are sequences such that $a_{n}>b_{n}\geq0$ for all $n\in\mathbb{N}$.
Further, suppose that $\sum_{k=1}^{\infty}\frac{1}{a_{k}}=\infty$ and $\sum_{k=1}^{\infty}b_{k}<\infty$.
Prove or disprove $\prod_{k=1}^{\infty}\big(1-\frac{b_{k}}{a_{k}}\big)>0$.

Thank you.
bkarpuz
$a_{n}:=\frac{1}{n}$ and $b_{n}:=\frac{1}{n^{2}}$ disproves the claim.
 
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