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?