Mathematics

Problem

What is the maximum possible length of the vector resulting from the following linear combination?

\frac{1}{\|\mathbf{v}_1\|} \mathbf{v}_1 + \frac{1}{\|\mathbf{v}_2\|} \mathbf{v}_2 + \cdots + \frac{1}{\|\mathbf{v}_n\|} \mathbf{v}_n

Hint(s): What is the length of each vector we're adding here? Think geometrically and try small cases -- if

n = 2

, what is the best you can do?