An infinite sequence is an ordered list of numbers, whereas an infinite series is the sum of the terms in an infinite sequence.
Let's break down the difference in more detail:
Infinite Sequence
- An infinite sequence is simply a list of numbers that continues indefinitely.
- Each number in the sequence is called a term.
- Sequences are often denoted as: a1, a2, a3, ..., an, ...
- Example: 1, 2, 3, 4, 5, ... is an infinite sequence of natural numbers.
Infinite Series
- An infinite series can be thought of as the sum of an infinite sequence. (Reference: 21-Aug-2014)
- It is represented as the sum of all the terms in an infinite sequence.
- Series are often denoted as: a1 + a2 + a3 + ... + an + ... or using summation notation: ∑ an (where the summation is from n=1 to infinity).
- Example: 1 + 2 + 3 + 4 + 5 + ... is an infinite series, representing the sum of the natural numbers.
Key Differences Summarized
| Feature | Infinite Sequence | Infinite Series |
|---|---|---|
| Definition | Ordered list of numbers | Sum of the terms in an infinite sequence |
| Representation | a1, a2, a3,... | a1 + a2 + a3 +... or ∑ an |
| Outcome | A list of numbers | A single value (if the series converges) or divergence |
| Example | 1, 4, 9, 16, 25, ... | 1 + 4 + 9 + 16 + 25 + ... |
In essence, a sequence is a list, and a series is the sum of the elements in that list. A series can converge to a finite value, meaning the sum approaches a specific number as more terms are added, or it can diverge, meaning the sum grows without bound.
