Sequences

For #1, I think you mistook it for a series instead of a sequence? I'm still on sequences. Before I started finding the limits, I tried to find a[SIZE="1"]1[/SIZE], a[SIZE="1"]2[/SIZE], a[SIZE="1"]3[/SIZE], etc.

A sequence is just a set of numbers. For example, {1, 2, 3, 4}. You're not really doing anything to them. A series implies that you're adding all the terms in a sequence together, and that's exactly what you're doing here.

4. Find the least upper bound and the greatest lower bound of a[SIZE="1"]n[/SIZE] = (e^n)/ (n+1)!
The sequence is decreasing, so I think the least upper bound would simply be a[SIZE="1"]1[/SIZE]. I don't know what to do for the lower bound.

Upper and lower bound is codeword for: check the end behavior.

lim e^n / (n+1)! as n approaches ∞

It's not immediately obvious to see this limit, but you can compare it to:

lim e^n / (n+1)^n = 0 as n approaches ∞

because e^n / (n+1)^n is always going to be greater than e^n / (n+1)! . Therefore, by comparison, lim e^n / (n+1)! is also 0 as n approaches ∞, so the lower bound is 0.