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Can we construct a sequence $\{a_{i}\}$, where $0 for all $i$, such that $\sum_{i=1}^{n}a_{i}=\frac{B_{n}}{2^{n}}$ such that $B_{n}\to b$ as $n\to \infty$ for some $1\leq b <\infty$?

Edit: What about $\sum_{i=1}^{n}a_{i}=\frac{n}{n+1}$, or $\sum_{i=1}^{n}a_{i}=B_{n}$ with $B_{n}\to b$, could we find such $a_{i}$?

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    @did: yes, this is working, by taking $a_{i}=\frac{1}{i(i+1)}$, Thanks!2012-06-24

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Whether $(B_n)$ converges or not does not really matter. That such a sequence exists is equivalent to $0<\underbrace{\frac{B_{n+1}}{2^{n+1}}-\frac{B_n}{2^n}}_{\text{ this would be }a_{n+1}}\leq 1,$ for $n\geq 1$, since this uniquely determines the sequence $(a_n)$ as above with $a_1=B_1/2$.

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    @did: You are right, the only nonnegative sequence for which this works is constant zero. It should work for some nonpositive sequences though.2012-06-24