Limit of Sequences

Bridge Course, August 2012

**Definition**: Consider an infinite sequence of numbers a_{1}, a_{2}, . . . , a_{n}, . . .. In many real-world situations, it tends to approach a constant number. We discuss rigorously the idea of a sequence converging towards a point called the limit. A number A is the limit of the sequence if the following is true: for any positive ϵ > 0, there exists an integer M (usually depends on ϵ) such that for all n > M,

.

We write it as

.

**Intuition**: Suppose we have a sequence of points (i.e. an infinite set of points labeled using the natural numbers), and has a concept of closeness (such as "all points within a given distance of a fixed point"). A point A is the limit of the sequence if for any prescribed closeness, all but a finite number of points in the sequence are that close to A. This may be visualized as a set of spheres of size decreasing to zero, all with the same center A, and for any one of these spheres, only a finite number of points in the sequence being outside the sphere.

Figure 1: Convergence of an infinite sequence.

**Properties**

1. If the limit of a sequence exists, then it is unique.

2. Every convergent sequence is bounded.

3. Every unbounded sequence is divergent.

4. If every a_{n} is in the domain of a function f and if f is continuous at each an, then.

If and , then

- , where is a constant
- (if and for all )

**Examples** **MAXIMA Code**

** **

- limit(1/n, n, inf)
- limit((a*n+1)/n, n, inf)
- limit(1+(-1)^n/n, n, inf)
- limit((-1)^n/n, n, inf)
- limit(1/n^2, n, inf)

**When limits do not exist **

The above graph shows that the limit of oscillates between 1 and -1 when n tends to infinity. Thus, does not exist.

**Examples**

1. Discuss the convergence of the sequence , where a is a real number.

(a)For , we let , where is a positive real number.

so

(b)For , we have

(c)For , we have , and by (a),

(d)For , we have

(e)For , we have

does not exist.

To conclude,

2. Find limit(n-3*n^2/(3*n+2), n, inf)

3. Find by using limit((1+1/(n-1))^n, n, inf)

**Exercises **

- Compute limit((n^3+100*n^2)/(4*n^3+5*n^2-1), n, inf)
- Compute ﬁnd it yourself
- Compute limit((4*n^2)/sqrt(16*n^4+1), n, inf)
- Compute ﬁnd it yourself
- Compute ﬁnd it yourself
- Compute ﬁnd it yourself
- Compute ﬁnd it yourself
- Show that if the limit of a sequence exists, then it is unique.
- (difficult) Show that .Use this and other results you learned in the properties section, compute .

## Comments (1)

## NG SIU KEI said

at 11:53 am on Aug 12, 2012

good

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