A bit rusty on this stuff. The whole problem is proving this is true:
$$ 2 \leq 1 + \sum_{m=1}^{n} \frac{1}{m!} \leq 1 + \sum_{m=1}^{n} \frac{1}{2^{m-1}} < 3. $$
I have figured out the first two inequalities:
$$2 \leq 1 + \sum_{m=1}^{n} \frac{1}{m!}, \quad \quad \text{and}$$ $$1 + \sum_{m=1}^{n} \frac{1}{m!} \leq 1 + \sum_{m=1}^{n} \frac{1}{2^{m-1}} .$$
But I am having trouble proving the last bit:
$$1 + \sum_{m=1}^{n} \frac{1}{2^{m-1}} < 3 .$$
Any tips?