Sequences and limits




Sequences and limit

Examples

The elements of a sequence are called the terms.
The 'n-th term' or 'general term' of the first example is (2n + 1).
The sequence is completely determined by this general term. Therefore we write the first sequence as {2n + 1}.
The second sequence is {1.2.3.4...n} or {n!}.
 
The third is

       {(-1)n+1 }
We write a general theoretical sequence as t1,t2,t3,... or {tn}.

Distance between two real numbers.

We define the distance between two real numbers a and b as |b - a|.

Base environment of a real number b.

Take a fixed real number b. For each strictly positive real number e, we say that the set of numbers { x | with b - e < x < b + e } is a base environment of b with radius e. We write this environment as ]b - e , b + e[. In most applications e is a very small strictly positive real number.

A finite limit

From this we see that

Criterion of Cauchy

tr criterion-of-cauchy cauchy-criterion Theorem:
 
   The sequence {tn} converges

                   <=>

   With each strictly positive real number e,
   corresponds a suitable positive integer N
    such that  n > N  => |tn+p - tn|< e  for each p=1,2,3,...
Proof:

First part: suppose the sequence {tn} converges to a number b.

A base environment of b, contains all the terms of {tn}, starting from a suitable term.
From this it follows that

 
   With each strictly positive real number e,
   corresponds a suitable positive integer N
    such that  n > N  => |tn+p - tn|< e  for each p=1,2,3,...
Second part: suppose that
 
   With each strictly positive real number e,
   corresponds a suitable positive integer N
    such that  n > N  => |tn+p - tn|< e  for each p=1,2,3,...
Choose a fixed value of n. We divide all real numbers in two sets.
A real number belongs to the set A if it is exceeded by an infinitely number of elements of the sequence.
A real number belongs to the set B if it is exceeded by a finite number of elements of the sequence.
Both sets define a Dedekind-cut b.

From |tn+p - tn|< e, we see that the number tn -e is exceeded by an infinitely number of elements of the sequence and belongs to set A.

From |tn+p - tn|< e, we see that the number tn +e is exceeded by a finite number of elements of the sequence and belongs to set B.

Therefore |tn - b| =< e.

 
Now, tn+p - b = tn+p - tn + tn - b

=> |tn+p - b| =< |tn+p - tn| + |tn - b| < 2e  for p=1,2,3,...

=> | tm - b | < 2e for all m > n

=> The sequence {tn} converges

Zero-sequence

Each sequence that converges to 0 is a zero-sequence . If for each n: tn>0 and lim tn=0, then we can write lim (tn) = +0.
If for each n: tn<0 and lim tn=0, then we can write lim (tn) = -0.

An infinite limit

sequences without a limit.

Not every sequence has a limit. Example: 1,-1,1,-1,1,-1,...

Bounded and monotone sequences

Bounded sequences ; monotone sequences

Take a sequence {tn}.
 
        A real number M is an upper bound for {tn}
                if and only if
                for each n : M >= tn

        A real number m is an lower bound for {tn}
                if and only if
                for each n : m  =< tn

                A sequence {tn} is bounded
                if and only if
        {tn} has an upper and a lower bound

If a sequence has a finite limit, then it is bounded.

Say lim tn = b, and choose a strictly positive number e. Then, only a finite number of term of the sequence are not in ]b - e,b + e[. Then it is always possible to choose an upper and a lower bound.
Remark: If a sequence is bounded, it has not always a finite limit.

Monotone sequences

If for all n tn+1 > tn then we say that the sequence is rising.
If for all n tn+1 < tn then we say that the sequence is descending.
If for all n tn+1 =< tn then we say that the sequence is not rising.
If for all n tn+1 >= tn then we say that the sequence is not descending.
If a sequence is not rising or not descending, we say that it is a monotone sequence .

Each Not descending and upper bounded sequence has a finite limit.

{tn} is a not empty upper bounded set; So, it has a smallest upper bound s. Then, there is a term tN in ]s - e , s + e[ for each strictly positive value of e. And since the sequence is not descending, all the following terms are in ]s - e , s + e[. Hence, lim tn = s.

Each not rising and lower bounded sequence has a finite limit.

The proof is analogous with the previous one.

Each not descending sequence, that hasn't an upper bound, diverges.

Choose a real number r. Since r is not an upper bound, there is a term tN > r and all following terms are > r . So, for each r, there is a N such that n > N => tn > r . Hence tn diverges.

Each not rising sequence, that hasn't a lower bound, diverges.

The proof is analogous with the previous one.

Properties of sequences

Without a proof we accept the following properties.
(Here n is a fixed integer.)

Rules for finite limits

Rules with infinity

We define the following rules for calculation rules with infinity.
We write infty for (+infty or -infty)
 
        -(+infty)=-infty  -(-infty)=+infty

        +(-infty)=-infty  +(+infty)=+infty

        |+infty|= +infty  |-infty|= +infty

        (+infty)n = +infty   (-infty)2n= +infty   (-infty)2n+1= -infty

        nth-root(+infty)=+infty   (2n+1)th-root(-infty)=-infty


        for each r = strictly positive real number

        r(+infty)=+infty  r(-infty)=-infty

        -r(+infty)=-infty   -r(-infty)=+infty


        for each  real number r

        r/+infty = 0    r/-infty = 0

        +infty + r = +infty    -infty + r = -infty

        r - infty = -infty    r + infty = +infty


        +infty +(+infty)= +infty

        -infty +(-infty)= -infty

        +infty -(-infty)= +infty

        -infty -(+infty)= -infty

        (+infty)(+infty)= +infty

        (-infty)(+infty)= -infty

        (+infty)(-infty)= -infty

        (-infty)(-infty)= +infty

Rules for infinite limits

 
        lim tn = +infty => lim (-tn) = -infty

        lim tn = -infty => lim (-tn) = +infty

        lim tn = +infty => lim |tn| = +infty

        lim tn = -infty => lim |tn| = +infty

if lim tn = +infty or lim tn = -infty, then

        lim tnp = (lim tn)p

        lim (nth-root(tn)) = nth-root(lim tn)  (if both sides exist)

        lim (c.tn) = c.(lim tn)       (with c real number)

        lim (c/tn) = 0         (with c real number)

        lim tn = +0 => lim (1/tn)=+infty

        lim tn = -0 => lim (1/tn)=-infty

if lim tn = infty and lim tn'= b (real and not zero) then

        lim(tn+tn')=lim tn +lim tn'

        lim(tn-tn')=lim tn -lim tn'

        lim(tn'-tn)=lim tn' -lim tn

        lim(tn'.tn)=lim tn' .lim tn

        lim(tn/tn')=lim tn / lim tn'

        lim(tn'/tn)= 0

        lim tn = +infty and lim tn' = +infty

                => lim(tn+tn')=lim tn + lim tn'

        lim tn = -infty and lim tn' = -infty

                => lim(tn+tn')=lim tn + lim tn'

        lim tn = +infty and lim tn' = -infty

                => lim(tn-tn')=lim tn - lim tn'

        lim tn = -infty and lim tn' = +infty

                => lim(tn-tn')=lim tn - lim tn'

        lim tn = infty and lim tn' = infty

                => lim(tn.tn')=lim tn . lim tn'

Arithmetic and Geometric sequences

About arithmetic sequences

Construction of an arithmetic sequence

Take a constant real number v, and define a sequence
 
        tn = t1 + (n-1).v
With each choice of t1 corresponds exactly one sequence .
All this sequences are called arithmetic sequences.
v is called the common difference of the arithmetic sequence.
If v = 0 the sequence is constant.
If v > 0 the sequence is rising and has no upper bound. lim tn = +infty.
If v < 0 the sequence is descending and has no lower bound. lim tn = -infty.

Sum of terms and arithmetic sequence.

Say S = t1 + t2 + ... + tn , then
 
S = t1 + t1 + v + t1 + 2.v + ... + tn

Now write the same sequence in reverse order

S = tn + tn - v + tn - 2.v + ... + t1

Addition gives

2.S = (t1 + tn).n

So,
            (t1 + tn).n
        S = ----------------
                  2

Application

Find the m-values such that the roots of
x4 - (1+3m) x2 + m2 = 0
form an arithmetic sequence.

The sum of the four roots is zero. So, we can write the roots as ( -3h, -h, h, 3h ). Then we have

 
   (x+3h)(x-h)(x+h)(x-3h) = x4 - (1+3m) x2  + m2
<=>
   (x4 - 9h2)(x2-h2) =  x4 - (1+3m) x2  + m2
<=>
   9h4 = m2
   10h2 = 1 + 3m

Case 1 :
   m = 3h2
   h2 = 1
   m = 3

Case 2 :
   m = -3h2
   19h2 = 1
   m = -3/19

About geometric sequences

Construction of a geometric sequence

Take a t1 and a constant real number q, and define a sequence
 
                tn = t(n-1).q
With each choice of t1 corresponds exactly one sequence .
All this sequences are called geometric sequences.
q is called the common ratio of the geometric sequence.

Theorem:

 
for all n > 1  :
If q = 1 + x > 1 , then qn  > 1 + n.x    (1)
Proof:

Corollary

Geometric sequences and limit

 
tn = t1.qn-1     = t1.qn /q = (constant number) .qn

If q > 1  then lim tn = + infty or -infty

If q = 1  then the sequence is constant lim tn = t1

If 0 < q <1  then  lim tn = 0

If -1< q <0  then  lim tn = 0

If q = -1  then lim tn don't exist

If q < -1 then lim tn don't exist

Sum of the first n terms of an geometric sequence.

 

        S   = t1 + t2 + ... + tn , then
=>      S.q = t1.q + t2.q + ... + tn.q
=>      S.q = t2 + ... + tn + tn.q

=>      S.q - S = tn.q - t1
=>      S(q-1) = tn.q - t1

             tn.q - t1          t1.qn - t1
=>      S = ---------------- = ----------------
               (q - 1)              (q - 1)

             t1.(qn - 1)
=>      S = ----------------
               (q - 1)

The sum of all terms of a converging geometric sequence.

 
Take 0 < |q| < 1

                 t1.qn - t1           t1
        S = lim ---------------- = -----------
                   (q - 1)             (1 - q)

Calculation of other sums

Sometimes it is possible to calculate the sum of n terms of a non-geometric row, using the properties of a geometric row.

We give an example. Say you want to calculate the sum

S = 1.2-1 + 2.2-2 + 3.2-3 + ... + n.2-n

We write S as follows:

 
2-1
2-2 + 2-2
2-3 + 2-3 + 2-3
2-4 + 2-4 + 2-4 + 2-4

. . . . .

2-n + 2-n + 2-n + 2-n + ... +2-n
Each column is the sum of terms of a geometric sequence with ratio 1/2.
The number of terms in a column starts with n and reduces to 1.
There are n columns. We calculate the sum of each column with the formula from geometric sequences.
 
The sum of the first column is, after simplification, 1  - 2-n
The sum of the second column is, after simplification,  2-1 - 2-n
The sum of the third  column is, after simplification,  2-2 - 2-n
The sum of the fourth  column is, after simplification,  2-3 - 2-n
......
The sum of the n-th  column is, after simplification, 21-n - 2-n
S = ( 1 + 2-1 + 2-2 + 2-3 + ... + 21-n ) - n.2-n

The first part is again a sum of n terms of a geometric sequence. We find after simplification:

S = 2 - 21-n - n 2-n

This was the ultimate goal. The sum of n terms of a non-geometric row was calculated using the properties of a geometric sequence.

Solved Problems

 
You can find solved problems about sequences using this link
 





Topics and Problems

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