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1 limit of sequences

Page history last edited by CAI, Sheng 11 years, 10 months ago

Limit of Sequences
Bridge Course, August 2012

 

     Definition: Consider an infinite sequence of numbers a1, a2, . . . , an, . . .. 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,
                                                                      Formula.
We write it as
                                                                 Formula.

 

     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 an is in the domain of a function f and if f is continuous at each an, thenFormula.

 

If Formula and Formula, then

  • Formula, where Formula is a constant
  • Formula
  • Formula
  • Formula(if Formula and Formula for all Formula)

 

Examples                                                                                                       MAXIMA Code

  

  1. Formula                                                                                  limit(1/n, n, inf)
  2. Formula                                                                           limit((a*n+1)/n, n, inf)
  3. Formula                                                                    limit(1+(-1)^n/n, n, inf)
  4. Formula                                                                           limit((-1)^n/n, n, inf)
  5. Formula                                    limit(1/n^2, n, inf)

 

When limits do not exist 

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

 

 

Examples

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

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

                                   Formula

                                         Formula

                                         Formula 

so Formula 

 

     (b)For Formula, we have Formula 

 

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

               Formula

 

     (d)For Formula, we have Formula

 

     (e)For Formula, we have Formula

          Formula does not exist.

To conclude, Formula

 

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

          Formula

                                                       Formula

                                                       Formula

                                                       Formula

 

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

 

          Formula

                                                       Formula

 

Exercises 

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

 

Comments (1)

NG SIU KEI said

at 11:53 am on Aug 12, 2012

good

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