let $a_n<c_n$ $\sum_n^\infty a_n=A$ and $\sum_n^\infty c_n=C$ $B\in(A,C)$ construct b_n such that $\sum_n^\infty b_n=B$

Question

I find this question tricky. I do not know how to approach a problem like this. I took a look at Direct comparison test proof, but it was not helpful, because no series were constructed there.

So what I want is a hint where to start.

Answer 1

Here's a hint:
$$B=\frac{B-A}{C-A}C+\frac{C-B}{C-A}A.$$