\frac{1}{2^k + 1}, \ldots, \frac{1}{2^{k + 1}} \geq \frac{1}{2^{k + 1}} \Rightarrow \frac{1}{2^k + 1} + \ldots + \frac{1}{2^{k + 1}} \geq 2^k \cdot \frac{1}{2^{k + 1}} = \frac{1}{2}