Section 1 A sequence(of real numbers) is  a list of infinitely many real numbers arranged in such a manner that it has a beginning but no end such as:  S 1 : 1, 2, 4, 8, …  S 2 : 1, ½, 1/3, ¼, 1/5, …  S 3 : 1, -1, 1, -1, …  S 4 : 1, -1/2, 1/3, -1/4, …  S 5 : 2, 2, 2, …  S 6 : -20, -10, -5, -4, 2, 2, 2, 2, … Increasing sequence Decreasing sequence Oscillating sequence Constant sequence Constant sequence! Among them, which are convergent sequences?

Convergent sequences  A sequence x n is said to be convergent  iff  (or we say that x n converges)  Otherwise, it is said to be divergent.  (or we say that x n diverges)

Sequence1 Is x n convergent? L= 0 ! Discussion:p.293 Ex.7.1, Q.2

Sequence 2: Does it converge? Does it converge? Discussion:Ex.7.1, Q.3

Two important sequences 1) Let q be a fixed real number and |q| < 1, then 2) Let a be a fixed positive number, then

Two important theorems

Section 2 Infinity  Which of the following sequences are divergent? How many categories are there? a) 0, 1, 0, 2, 0, 3, 0, 4, … b) 2/1, 4/2, 8/3, 16/4, … c) 1, -1, 2, -2, 3, -3, 4, -4,… d) x n = n Oscillating Tends to infinity Discussion : p.298 Ex.7.2, 4-6

Section 3 Bounded and unbounded sequences We say that Xn is bounded by 1.5 since |x n |< 1.5 for any natural no. n.

Is bounded?

Bounded above and below  is bounded below by 0.  is bounded above by 9.  is both bounded above and below. i.e. it is bounded. x n > 0 for all n. x n < 9 for all n. |x n |<1 for all n.

An important theorem  Can a convergent sequence be unbounded?  If it approaches to L as n tends to infinity, then it can’t go too far from L.  Therefore every convergent sequence must be bounded. Discussion : Ex.7.3 Q.2

Section 4 Properties of a sequence  Theorem 4.1

The uniqueness of limit  The limit of a convergent sequence is unique.  Reason: The sequence can’t have two ‘continuous’ tails.

Sandwich Theorem Can you state the theorem? (Principle of Squeezing, or Squeezing Theorem)

Statement of Sandwich Theorem Example 4.1

Example1: Zn=5.2 1/n y n =(4 n +5 n ) 1/n Xn =5

Prove that

Prove that is convergent. This is a very common mistake since limits can’t be evaluated by splitting into infinite many pieces though each of them is convergent!

Prove that is convergent.

For {x n /y n }, y n are non-zero and lim y n are non-zero too. For {x n /y n }, y n are non-zero and lim y n are non-zero too. Section 5 Operations of Limits of Sequences

Three important theorems

Proof:

Theorem 5.8  Theorem 5.8 Is the converse correct? Counter-example: {(-1) n } Can the converse be true for some value(s) of L?

Theorem 5.9

Section 8 Monotonic Sequences  Theorem 8.1 If a sequence is monotonic increasing increasing(decreasing) and is bounded above(below), then it is convergent i.e. it has a limit.

Example 8.1  Show that the sequence is convergent and find its limit.

Proof of example 8.1

AA problem for discussionbc

Example 8.3 Discussion on Ex.7.5 Q.5