Sequences and limits

Themes > Science > Mathematics > Algebra > Foci of a conic section > Topics and Problems > Sequences and limits

..Examples
..Distance between two real numbers.
..Base environment of a real number b.
..A finite limit
..Criterion of Cauchy
..Zero-sequence
..An infinite limit
..sequences without a limit.

..Bounded and monotone sequences

..Bounded sequences ; monotone sequences
..If a sequence has a finite limit, then it is bounded.
..Monotone sequences
..Each Not descending and upper bounded sequence has a finite limit.
..Each not rising and lower bounded sequence has a finite limit.
..Each not descending sequence, that hasn't an upper bound, diverges.
..Each not rising sequence, that hasn't a lower bound, diverges.

..Properties of sequences

..Rules for finite limits

..Rules with infinity

..Rules for infinite limits

..Arithmetic and Geometric sequences

..About arithmetic sequences

..Construction of an arithmetic sequence
..Sum of terms and arithmetic sequence.

..About geometric sequences

..Construction of a geometric sequence
..Theorem:
..Corollary
..Geometric sequences and limit
..Sum of the first n terms of an geometric sequence.
..The sum of all terms of a converging geometric sequence.

Sequences and limit

Examples

3,5,7,9,11,...
1,2,6,24,120, ...
1,-1,1,-1,1,...

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 {t_n}.

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

Example : Take the sequence 1/2 , 2/3 , 3/4 , 4/5 ,...
If n grows, the n-th term is almost 1. The distance between the n-th term and 1 can become arbitrary small if n is very large.
With each strictly positive real number e, corresponds a suitable positive integer N such that
```
 
        n > N  => |t_n - 1|< e
```
We say that the limit of t_n = 1 and we write lim t_n = 1.

General definition : Take the sequence {t_n}. We say that

 
        lim t_n = b

           <=>
   With each strictly positive real number e,
   corresponds a suitable positive integer N
   such that  n > N  => |t_n - b|< e .

We say that the sequence converges to b.

From this we see that

If a sequence converges to b, then a base environment of b, contains all the terms of {t_n}, starting from a suitable term.
If each base environment of b contains all the terms of {t_n}, starting from a suitable term, then the sequence converges to b.

Criterion of Cauchy

tr criterion-of-cauchy cauchy-criterion Theorem:

 
   The sequence {t_n} converges

                   <=>

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

Proof:

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

A base environment of b, contains all the terms of {t_n}, 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  => |t_n+p - t_n|< 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  => |t_n+p - t_n|< 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 |t_n+p - t_n|< e, we see that the number t_n -e is exceeded by an infinitely number of elements of the sequence and belongs to set A.

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

Therefore |t_n - b| =< e.

 
Now, t_n+p - b = t_n+p - t_n + t_n - b

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

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

=> The sequence {t_n} converges

Zero-sequence

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

An infinite limit

Example : Take the sequence {n.n} . The general term t_n can grow bigger than any real number r.
With each real number r, corresponds a suitable positive integer N such that
```
 
        n > N  => t_n > r
```
We say that t_n has an infinite limit and we write lim t_n = infinity.

General definition : Take the sequence {t_n}. We say that

 
        lim t_n = +infty

           <=>
   With each real number r,
   corresponds a suitable positive integer N
   such that  n > N  => t_n > r .

        lim t_n = -infty

           <=>
   With each real number r,
   corresponds a suitable positive integer N
   such that  n > N  => t_n < r .

We say that the sequence diverges to infinity.

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 {t_n}.

 
        A real number M is an upper bound for {t_n}
                if and only if
                for each n : M >= t_n

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

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

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

Say lim t_n = 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 t_n+1 > t_n then we say that the sequence is rising.
If for all n t_n+1 < t_n then we say that the sequence is descending.
If for all n t_n+1 =< t_n then we say that the sequence is not rising.
If for all n t_n+1 >= t_n 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.

{t_n} is a not empty upper bounded set; So, it has a smallest upper bound s. Then, there is a term t_N 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 t_n = 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 t_N > r and all following terms are > r . So, for each r, there is a N such that n > N => t_n > r . Hence t_n 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.)

 
If lim t_n = b and lim t_n' = b
and if   for all n > N : t_n =< t"_n =< t_n'
Then lim t"_n = b

 
If lim t_n = b
and if   for all n > N : t_n = t_n'
Then lim t_n' = b

 
If lim t_n = +infty
and if   for all n > N : t_n =<  t_n'
Then lim t_n' = +infty

 
If lim t_n = -infty
and if   for all n > N : t_n >=  t_n'
Then lim t_n' = -infty

 
If lim t_n = b > 0
Then t_n > 0 for all n starting from a suitable n=N

 
If lim t_n = b < 0
Then t_n < 0 for all n starting from a suitable n=N

```
 
If lim t_n = b
Then lim |t_n| = |b|
```

Rules for finite limits

 
If lim t_n = b is a real number
Then,
        lim r.t_n = r.lim t_n

        lim 1/t_n = 1/lim t_n    (if b not 0)

        lim t_nⁿ  = ( lim t_n )ⁿ

        lim t_n^1/p = ( lim t_n )^1/n       (if both sides exist)

 
If lim t_n and lim t_n' are real numbers,
Then,   lim (t_n + t_n') = lim t_n + lim t_n'

        lim (t_n - t_n') = lim t_n - lim t_n'

        lim t_n.t_n' = lim t_n . lim t_n'

            t_n       lim t_n
        lim -----  =  ------------    (if both sides exist)
            t_n'      lim t_n'

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

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

        lim t_n = +infty => lim |t_n| = +infty

        lim t_n = -infty => lim |t_n| = +infty

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

        lim t_n^p = (lim t_n)^p

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

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

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

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

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

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

        lim(t_n+t_n')=lim t_n +lim t_n'

        lim(t_n-t_n')=lim t_n -lim t_n'

        lim(t_n'-t_n)=lim t_n' -lim t_n

        lim(t_n'.t_n)=lim t_n' .lim t_n

        lim(t_n/t_n')=lim t_n / lim t_n'

        lim(t_n'/t_n)= 0

        lim t_n = +infty and lim t_n' = +infty

                => lim(t_n+t_n')=lim t_n + lim t_n'

        lim t_n = -infty and lim t_n' = -infty

                => lim(t_n+t_n')=lim t_n + lim t_n'

        lim t_n = +infty and lim t_n' = -infty

                => lim(t_n-t_n')=lim t_n - lim t_n'

        lim t_n = -infty and lim t_n' = +infty

                => lim(t_n-t_n')=lim t_n - lim t_n'

        lim t_n = infty and lim t_n' = infty

                => lim(t_n.t_n')=lim t_n . lim t_n'

Arithmetic and Geometric sequences

About arithmetic sequences

Construction of an arithmetic sequence

Take a constant real number v, and define a sequence

 
        t_n = 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 t_n = +infty.
If v < 0 the sequence is descending and has no lower bound. lim t_n = -infty.

Sum of terms and arithmetic sequence.

Say S = t₁ + t₂ + ... + t_n , then

 
S = t₁ + t₁ + v + t₁ + 2.v + ... + t_n

Now write the same sequence in reverse order

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

Addition gives

2.S = (t₁ + t_n).n

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

About geometric sequences

Construction of a geometric sequence

Take a t₁ and a constant real number q, and define a sequence

 
                t_n = 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)

Prove:

 
The theorem holds for n = 2.            (2)

Now suppose the theorem holds for n = k.
We'll prove that it holds for n = k+1.

 
         q^k  > 1 + k.x

=>      q.q^k  > (1+x).(1 + k.x)

=>      q^k+1  > 1 + (k + 1)x + k.x.x

=>      q^k+1  > 1 + (k + 1)x                 (3)

From (2) and (3) it follows that (1) holds for all n >1.

Corollary

From this theorem it follows that

 
If q > 1 then qⁿ  is rising and has no upper bound. lim qⁿ  = +infty

If 0 < |q| < 1 then

 

          1                  1  n
        (---) > 1  and lim (---)  = +infty
         |q|                |q|

Hence,
              n           1          1
        lim |q | = lim -------- = --------- = 0
                         1  n     +infty
                       (---)
                        |q|

Geometric sequences and limit

 
t_n = t₁.q^n-1     = t₁.qⁿ /q = (constant number) .qⁿ

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

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

If 0 < q <1  then  lim t_n = 0

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

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

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

Sum of the first n terms of an geometric sequence.

 

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

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

             t_n.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)

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