Let us consider two sequences of real numbers $a\_n$ and $b\_n$, about which we only know that: 
$$\sum\_{1}^{\infty}a\_n = 0$$ 
and that all $b\_n > 0$, with $b\_{n+1} > b\_n$. Can it be proved that there cannot exist a $b\_n$ sequence with said features, such that also
$$\sum\_{1}^{\infty}a\_n b\_n = 0 \;\;?$$
or is such hypothesis unjustified, and there instead exist counterexamples ?


Thanks


Following Mark's reply, I am reformulating into a somewhat different question, which is also of interest to me. 


Given the above conditions for $a\_n$ and $b\_n$, would it be possible to find additional general conditions for the $a\_n$'s which will then be sufficient to ensure that
$$\sum\_{1}^{\infty}a\_n b\_n \neq 0 \;\;?$$



On the series of the product of the terms of two sequences whose respective series are one convergent and the other not
Consider the series $a=1-1+1/4-1/4+1/9-1/9...+1/n^2-1/n^2...$ and $b=1+1+2+2+3+3+4+4...+n+n...$ . Their "product" is $1-1+1/2-1/2....1/n-1/n...$ converges to 0.If you really want $b\_{n+1}\gt b\_n$, then instead $...n+n...$ consider the series $...n+(n+1/n^2)...$.


