2.

(a) What is a convergent sequence? Give two examples.

(b) What is a divergent sequence? Give two examples.

(a)

To define: A convergent sequence with two examples.

Definition used:

If a sequence {an} has a limit l, then the sequence is convergent sequence, which can be written as limn→∞an=l . That is, limn→∞an exists.

Examples:

Examples of convergent sequence may vary. The possible two examples are as follows:

(1) The sequence {1n} is a convergent sequence, which converges to 0.

(2) The sequence {nn+1} is a convergent sequence, which converges to 1.

(b)

To define: A divergent sequence with two examples.

If a sequence {an} has no limit, then the sequence is termed as divergent sequence. That is, limn→∞an does not exist.

Examples of divergent sequence may vary; the possible two examples are as follows:

(1) The sequence {n8+n} is a divergent sequence as limn→∞{n8+n}=∞ .

(2) The sequence {sinn} is a convergent sequence as limn→∞{sinn} does not exist.