notes/education/math/MATH1220 (calc II)/Sequences.md

A sequence is defined as an ordered list of numbers.

• Sequences are ordered, meaning two sequences that contain the same values but in a different order are not equal.
• Sequences can be infinite if a rule is defined, i.e \{1, 1, 1, 1, ...\}; a_i = 1

Behavior

• A sequence is considered increasing if a_n is smaller than a_{n+1} for all n.

• A sequence is considered decreasing if a_n is greater than or equal to a_{n+1} for all n.

• Sequences exist that do not fall into either category, i.e, a_n = (-1)^n

• If the terms of a sequence grow \{a_n\} get arbitrarily close to a single number L as n grows larger, this is noted by writing: \lim_{n\to\infty} a_n = L OR

a_n \to L \text{ as } n \to \infty

and say that a_n converges to L. If no L exists, we say \{a_n\} diverges.

Properties of Sequences

The below properties assume two sequences are defined, a_n \to L and b_n \to M

1. a_n + b_n \to L + M
2. C*a_n \to CL
3. a_n b_n \to LM
4. \frac{a_n}{b_n} \to \frac{L}{M} holds true where all values are defined
5. If L = M and a sequence c_n exists such that a_n \le c_n \le b_n for all n, then c_n \to L = M
6. If a_n and b_n both approach infinity at a similar rate, \frac{a_n}{b_n} will approach an arbitrary value. This value can be found by rewriting \frac{a_n}{b_n} in such a manner that the end behavior of the series is more easily identifiable

   For example, given the series c_n = \frac{n}{2n+1}, both the numerator and the denominator approach infinity