Sequences and limits

Sequences and limit
Bounded and monotone sequences
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

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)

You can find solved problems about sequences here

Topics and Problems

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