Differentiate between mathematical series and a sequence. Give examples.

A sequence is a list of numbers arranged in a specific order, while a series is the sum of the terms in a sequence.

Example of a sequence: 1, 2, 3, 4, 5, ...

Example of a series: 1 + 2 + 3 + 4 + 5 + ... = 15

In the sequence example, the numbers are listed in order, while in the series example, the terms of the sequence are added together to get a sum.

Sequence:

- An ordered list of numbers or terms arranged in a specific pattern.
- Each term is called an element or a member of the sequence.
- The order of the terms is significant.
- Example: (2, 4, 6, 8, 10) is a sequence of even numbers.

Mathematical Series:

- A sum of an infinite number of terms of a sequence.
- The terms are added together to form a single value.
- The order of the terms in the series is not essential.
- Example: 1 + 1/2 + 1/4 + 1/8 + ... (sum of an infinite geometric series)

Key Differences:

- Finiteness: A sequence is finite (has a limited number of terms), while a series is typically infinite.
- Sum: A sequence does not have a sum, while a series represents the sum of its terms.
- Order of Terms: The order of terms is important in a sequence but not in a series.
- Notation: A sequence is typically written as {a1, a2, a3, ...} or (a_n), where n is the term number. A series is written as the sum of its terms, e.g., ∑(a_n).

Example:

Consider the sequence (1, 3, 5, 7, 9).

- The sum of the terms of this sequence is the series: 1 + 3 + 5 + 7 + 9.
- The sum of the series can be calculated as 25.