Can we construct a sequence $\{a_{i}\}$, where $0
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 $a_{i}$?
Can we construct a sequence $\{a_{i}\}$, where $0
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 $a_{i}$?
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 $a_1=B_1/2$.