Sequences, Series, and Limits

3.1 Definition of a Sequence

Definition 3.1: A sequence is a function whose domain is the positive integers (e.g., written f(n), n = 1, 2, 3, . . .).

Definition 3.2: A sequence has the limit L if, for any ϵ > 0, there is some value N such that |a_n − L| < ϵ whenever n > N.
A sequence with a limit is convergent, denoted as \lim{n→∞} an = L.
Sequences can be bounded or unbounded.
- A sequence is bounded if there's a finite value K > 0 such that for some N, an < K for all n > N (bounded above) and an > K for all n > N (bounded below).
Definition 3.3: If a sequence has no limit, it is divergent.
Definition 3.4: A divergent sequence is definitely divergent if:
- For any value K, there exists N such that an > K for all n > N, then \lim{n→∞} a_n = ∞.
- For any value K, there exists N such that an < −K for all n > N, then \lim{n→∞} a_n = −∞.

Present value calculation compares the value of money received at different times, putting them on equal footing with current money.
If a rate of return of 10% is possible, then $90.91 invested generates $100 at year-end; the present value of $100 to be received in one year at a 10% interest rate is $90.91.
Higher interest rates reduce the present value of future sums.
The general formula for present value (X) of V received in one year is X = \frac{V}{1 + r}, where r is the rate of return.