The answer is no. E.g., let $x:=(\frac12,\frac12,\frac13,\frac14,\ldots)$ and $y:=(0,\frac12,\frac13,\frac14,\ldots)$. Then $\|x+y\|=\|x\|+\|y\|$, but $x$ and $y$ are linearly independent.
However, it is rather easy to see that the answer will become yes if the summation $\sum_{i=2}^{\infty}$ in the definition of the norm is replaced by $\sum_{i=1}^{\infty}$.