Sequence and Limit

 

Motivation:

                     Let’s check the following :

 

                                    1, 3, 5, 7, 9, 11, 13, ….

 

                    What is the next number going to appear?  What’s the number at 101^st  place?

                    Another example is:

 

                                    9, 5, 3, 2, (1+½), (1+¼), (1+ 1/8 ),   …

 

                    We have similar questions: what is the next number going to appear?

                    Will  “-1”  appear in the sequence? 

                    Is there any “boundary” for the sequence ?

 

                    Mathematically, a sequence can be treated as a function whose domain is natural numbers. It is quite straightforward that you call something in the 1^st place, 2^nd place, 3^rd place, … Similarly in the math language, we can use

 

                                   a₁, a₂, a_3, a₄, …

 

                    And you will know which one is first, second, third, …

                    In general, we use   to represent a sequence. Or we simply write it as . 

 

Definition ( Sequence )

                  A sequence is a function whose domain is N.  We denote a sequence as  or   for short.

 

Example:  List the first 5 terms of the sequence  ,

                                   where  =2n + 1, n= 1, 2, 3, 4, 5, …

 

                Sol:

                             They are  3, 5, 7, 9, 11.

 

Example: List the 100th term of the sequence  { b_n },  where b_n = 2ⁿ, n=1,2,3,4,5, 6, …

 

              Sol:     

                              {b_n} is   2, 4, 8, 16, 32, 64, ..

                               The 100^th term is  2¹⁰⁰ .

 

 

 

Example:     where c_n =  1+   .

                   Find the first term such that  c_n -1 <  ½  .

 

          Sol:

                                             c_n -1 < ½

                                       1+  -1 < ½

                                        <

                                      2^(4-n) < 2^-1

                                    

                                      4-n < -1

                                             5 < n

                    Thus, when n=6, it is the first time that  c_n -1 <  ½ .

 

 

                    In the previous example, what if    c_n -1 <  0.01 ?  Or  c_n -1 <  0.0001 ? It seems that  the sequence  is approaching to 1.  It seems that no matter how small number is given,  there always exists an integer  k such that  (c_n – 1) is less than that small number when n is larger than k.  In other words, every term after  c_k is with that property.   Actually, this is how we define the limit of a sequence.

 

 

 

 

 

 

Infinity and Infinitesimal

 

                 How do we usually describe a very large number in our daily life?  Tons of money, a sea of cars, … Or when something is very good,  you might want to say “it is the best in the world” or “nothing in the world is better than that” .   Similarly,  we use this concept to describe “something” larger than any number.  Mathematically, we denote it as “  ” ( infinity ) .  And it has the property that it is larger than any number you specify.

 

            And when something is  very very small with negligible amount, we use an “operation model” to describe it :  no matter how small number you specify, the absolute value of this one is always smaller than the number you specify.  We bring the two concepts to have the definition for limit of a sequence.

 

 

 

 

 

 

 

 

 

Definition ( Limit of a Sequence )

             Let   be a sequence such that  c_n  R . If there exists L such that

 

                         for any  > 0, we can find a corresponding integer K and

                                       | c_n – L | <   when n > K

 

            Then we say that  the limit of the sequence  is  L.  It is denoted as

                                      

 

            That means: when n goes to infinity,  c_n is approaching to L . 

 

Note:             Sometimes,  you will see the notation as follows:

 

                                          > 0,         K such that

                                        | c_n – L | <   when n > K

 

                      “  > 0”  means “for any  >0” ;

                      “  K “   means “there exists  K “.

 

 

 

 

 

 

 

 

Example:   where c_n =   .   Then   .

        Proof:

                        To prove the limit of the sequence is 0, we need to follow the “operation model” of the definition.  Assume  is given, how can we find the corresponding K to satisfy the requirement to claim this limit is what we want?  We think in this way:

 

                                       | c_n -0 |  <

                                      <

                                       < n

 

              With this in mind, we start our verification.

               For any   > 0,   we can set K is equal to the least integer such that

                                             K > .   Then when n > K, we have

 

                                   | c_{n - 0} | =   <  <

              Thus,

                                   .

 

 

 

             The definition of  “Limit of Sequence” does not tell you how to effectively find the limit of a sequence.   It just tells you if you suspect the limit of the sequence is equal to L,  this L has to satisfy some properties.   But you can follow the example above to prove some simple things.

 

 

Exercise:  Prove