Key Features of Infinite Sequences

1. What is it?
   An infinite sequence is a list of numbers that goes on forever. We can write it like this: ( a_1, a_2, a_3, \ldots ).

2. Terms:
   Each number in the sequence is called a term. The spot of each term is shown by a whole number, like 1, 2, 3, and so on.

3. Types of Sequences:

   • Convergent Sequences: These sequences get closer and closer to a specific value, called a limit, as we look at more and more terms.
   • Divergent Sequences: These sequences do not settle down to any limit; they keep growing or changing forever.

4. Examples:

   • The sequence ( a_n = \frac{1}{n} ) gets really close to 0 as ( n ) gets larger.
   • The sequence ( b_n = n ) keeps getting bigger and goes to infinity.

5. How to Write Them:
   We can write infinite sequences using formulas. For instance, ( a_n = 2n + 1 ) gives us all the odd numbers.