How do you denote an infinite sequence?

An infinite sequence is typically denoted as a₁, a₂, ..., a_n, ... or {a_n}.

Here's a breakdown of what that means:

• a₁, a₂, ..., a_n, ...: This explicitly lists the first few terms of the sequence, followed by an ellipsis (...) to indicate that the sequence continues indefinitely according to a pattern, culminating in the general term a_n, and the final ellipsis showing the sequence is infinite.

• {a_n}: This is a more compact notation, representing the entire infinite sequence by its general term, a_n. It implies that 'n' takes on all positive integer values (n = 1, 2, 3, ...). In other words, the sequence is a function whose domain is the set of natural numbers (positive integers).

Examples:

1. Sequence of even numbers: 2, 4, 6, 8, ... or {2n}
2. Sequence of squares: 1, 4, 9, 16, ... or {n²}
3. Sequence of reciprocals: 1, 1/2, 1/3, 1/4, ... or {1/n}

In each of these examples, a_n represents the nth term of the sequence. The "... " signifies that the sequence continues infinitely.