Difference between Sequence and Series

Sequence

Concept: A sequence is simply an ordered list of numbers arranged according to a specific pattern or rule. Each number in the list is called a term.

Examples:

• Sequence of even numbers: $2, 4, 6, 8, \dots$
• Sequence of squares: $1, 4, 9, 16, \dots$

The focus is on each individual term and how the pattern forms.

Analogy: Think of a sequence like stairs. Each step is a term, and there's a clear order from the first step, second step, and so on.

Series

Concept: With a series, we sum up the terms of a sequence. So, a series is the result of the cumulative addition of the terms in a sequence.

Examples (using the sequences above):

• Series of even numbers: $2 + 4 + 6 + 8 + \dots$
• Series of squares: $1 + 4 + 9 + 16 + \dots$

The focus is on the total value or sum of the terms up to a certain $n$-th term (usually denoted as $S_n$).

Analogy: Back to the stairs analogy. If the sequence represents the individual steps, the series represents the total height you've climbed after taking a certain number of steps.

Core Difference