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, . . .).

3.2 Limit of a Sequence

• 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 $\lim<em>{n→∞} a</em>n = L$.
• Sequences can be bounded or unbounded.
  • A sequence is bounded if there's a finite value K > 0 such that for some N, $a<em>n < K$ 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 $a<em>n > K$ for all $n > N$, then $\lim</em>{n→∞} a_n = ∞$.
  • For any value K, there exists N such that $a<em>n < −K$ for all $n > N$, then $\lim</em>{n→∞} a_n = −∞$.

3.3 Present-Value Calculations

• 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.