# Finding all sequences of real numbers with a certain property

Let the series $\sum_{k \in N}a_k$ be convergent such that $\sum_{k \in N}a_k\neq 0$. Find all sequences $(b_k)_{k \in N}$ with the following properties: (i) the series $\sum_{k \in N}a_k b_k$ is convergent; (ii) $(\sum_{k \in N}a_k b_k)\times (\sum_{k \in N}a_k)>0$;

P.S. We assume that $(a_k)$ and $(b_k)$ are sequences of real numbers.

If $(b_k), (c_k)$ are such that they each $(b_k) \cdot \sum a_k = \sum (b_k a_k)$ and $(c_k) \cdot \sum a_k$ convergent then prove that $(b_k - c_k) \cdot \sum a_k$ is also convergent. If both $(b_k), (c_k)$ both share properties (i), (ii), then you can get the set of all such sequences to be closed under $(b_k + c_k)$ and to have an additive identity. So your sought-after set of sequences forms an additive monoid of real sequences.
• You also have closure under scalar multiplication by $\lambda \in \Bbb{R}_{\gt 0}$. So you have some type of semi-vector space. – StudySmarterNotHarder Jun 1 '14 at 0:29