Let $\left \{ a_{n} \right \}_{n=1}^{\infty }$ and $\left \{ b_{n} \right \}_{n=1}^{\infty }$ be two sequences of real numbers such that the series $\sum _{n=1}^{\infty }a_{n}^{2}$ and $\sum _{n=1}^{\infty }b_{n}^{2}$converge. Then the series $\sum _{n=1}^{\infty }a_{n}b_{n}$
- is absolutely convergent
- may not converge
- is always convergent, but may not converge absolutely
- converges to $0$