Precalculus linear combination problem What is the maximum possible length of the vector resulting from the following linear combination?
$$
\frac{1}{\| \mathbf{v_1} \|} \,\mathbf{v_1} + \frac{1}{\|\mathbf{v_2}\|} \,\mathbf{v_2} + \cdots + \frac{1}{\| \mathbf{v_n} \|} \,\mathbf{v_n}
$$

Let $\mathbf{u_i} = \frac{1}{\| \mathbf{v_i} \|} \,\mathbf{v_i}$, then $\| \mathbf{u_i} \| = 1.$
$
\Big\|\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}\Big\| = 
\| \mathbf{u_1} +  \mathbf{u_2} + \dots + \mathbf{u_n}\| \le \| \mathbf{u_1}\| +  \|\mathbf{u_2}\| + \dots + \|\mathbf{u_n}\| = n.$
