Sequences and limits

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  => |tn - 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 tn = b
  
             <=>
     With each strictly positive real number e,
     corresponds a suitable positive integer N
     such that  n > N  => |tn - 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 {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 {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  => |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 |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, 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: 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  => tn > 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 tn = +infty
  
             <=>
     With each real number r,
     corresponds a suitable positive integer N
     such that  n > N  => tn > r .
  
          lim tn = -infty
  
             <=>
     With each real number r,
     corresponds a suitable positive integer N
     such that  n > N  => tn < 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 {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 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 tn = b and lim tn' = b
  and if   for all n > N : tn =< t"_n =< tn'
  Then lim t"_n = b

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

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

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

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

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

• If lim tn = b
  Then lim |tn| = |b|

Rules for finite limits

• If lim tn = b is a real number
  Then,
          lim r.tn = r.lim tn
  
          lim 1/tn = 1/lim tn    (if b not 0)
  
          lim tnn  = ( lim tn )n
  
          lim tn1/p = ( lim tn )1/n (if both sides exist)

• If lim tn and lim tn' are real numbers,
  Then,   lim (tn + tn') = lim tn + lim tn'
  
          lim (tn - tn') = lim tn - lim tn'
  
          lim tn.tn' = lim tn . lim tn'
  
              tn       lim tn
          lim ----- = ------------ (if both sides exist)
              tn'      lim tn'

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 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 + ... + 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

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)

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.
  

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

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)