Mathematical Analysis

Sequences and Series

What is a sequence?

A sequence is defined as a set of real numbers with a natural order. A sequence is usually included in brackets ({}), with the terms, or parts of a sequence, separated by commas. For example, if a scientist collects weather data every day for many days, the first day of collecting can be written as x1 data; then x2 for the second day, and so on until xn, in which n is the eventual number of days. This can be written as {x1, x2, … xn} n≥1. In general, the sequence of numbers in which xn is the nth number is written using the following notation: {xn}n≥1.

A sequence can get larger or smaller. For example, in the sequence for {2n}n≥1, the solution is 2 ≤ 4 ≤ 8 ≤ 16 ≤ 32, and so on, with the numbers getting larger. Whereas, for {1/n}n≥1, the sequence becomes 1 ≥ ½ ≥ 1/3 ≥ 1/4 ≥ 1/5, and so on, with the numbers getting progressively smaller. This does not mean that sequences only get progressively larger and smaller; certain solutions for sequences include a mix of the two.


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