A sequence is defined as: a function whose domain is the set of positive integers.
Notation Entire sequence: $\left\{ a_n \right\}$ nth term: $a_n$
Basic Sequence The terms of the sequence $\left\{ a_n \right\} = \left\{ \frac{n^2}{2^n - 1} \right\}$ are $\left(\frac{1^2}{2^1 - 1}\right), \left(\frac{2^2}{2^2 - 1}\right), \left(\frac{3^2}{2^3 - 1}\right), \left(\frac{4^2}{2^4 - 1}\right), ...$
Recursive Sequence
e.g. $\left\{ d_n \right\}$ has $d_1 = 25$ and $d_{n+1} = d_n - 5$
The most important concept in learning about infinite sequences is sequences whose terms approach limiting values. Such a sequence converges.
Convergent Sequence This sequence converges to $0$: $$\left\{ \frac{1}{2^n} \right\} = \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, ...$$
Definition: Limit of a Sequence Let $L$ be a real number. The limit of a sequence $\left\{ a_n \right\}$ is $L$,written as
$$\lim_{n\to \infty} a_n = L$$if for each $\varepsilon > 0$, there exists $M > 0$ such that $| a_n - L | < \varepsilon$ whenever $n > M$.
If the limit $L$ of a sequence exists, then the sequence converges to $L$.
If the limit of a sequence does not exist, then the sequence diverges.
Because a sequence is a function whose domain is restricted to the set of all positive integers, the limit laws for sequences are very similar to those for functions.
Let $a_n$ and $b_n$ be sequences of real numbers, and let $A$ and $B$ be real numbers. The the following rules are true if:
$\lim_{n\to\infty}\{a_n\} = A$ and $\lim_{n\to\infty}\{b_n\} = B$
| __Sums__ | The limit of the sum is the sum of the limits: $$\lim_\limits{n\to\infty} (a_n + b_n) = A + B$$ |
| __Differences__ | The limit of the difference is the difference of the limits: $$\lim_\limits{n\to\infty} (a_n - b_n) = A - B$$ |
| __Constant Multiples__ | The limit of the constant multiple is the constant multiple of the limit: $$\lim_\limits{n\to\infty} (k \cdot a_n) = k \cdot A \text{ (for any number k)}$$ |
| __Products__ | The limit of the product is the product of the limit: $$\lim_\limits{n\to\infty} (a_n \cdot b_n) = A \cdot B$$ |
| __Quotients__ | The limit of the quotient is the quotient of the limit: $$\lim_\limits{n\to\infty} \left(\frac{a_n}{b_n}\right) = \frac{A}{B} \text{ if }B \ne 0$$ |