MT2002 Analysis
Previous page
(Axioms for the Real numbers) 		Contents Next page
(Some properties of convergent sequences)

Convergence in the Reals

One of the two most important ideas in Real analysis is that of convergence of a sequence.

It took mathematicians some time to settle on an appropriate definition. See Some definitions of the concept of a limit.

Definition

A sequence in R is a list or ordered set: (a1 , a2 , a3 , ... ) of real numbers.

Examples

One may define a sequence (an) by giving an explicit formula for the nth term.

  1. (1/n) = ( 1 , 1/2 , 1/3 , ... )

  2. (sin(<fontface=symbol>pn /4)) = (1/sqrt2 ,1 , 1/sqrt2 , 0 , -1/sqrt2 ,-1 , -1/sqrt2 , 0 , ... )

  3. ( (5n+2)/(3n+1) ) = ( 7/4 , 12/7 , 17/10 , 22/13 , 27/16 , ... )

One may define a sequence by a recurrence relation. This gives a formula for the nth term in terms of earlier ones.

  1. an+1 = an/(an+ 1), a1 = 1.
    This gives the sequence ( 1 , 1/2 , 1/3 , 1/4 , ... ).

  2. fn+2 = fn+1 + fn and f1 = f2 = 1.
    This gives (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... )

    This is the Fibonacci sequence first introduced by the Italian mathematician Fibonacci or Leonard of Pisa (1170 to 1250) and much studied by Edouard Lucas (1842 to 1891).

  3. an+1 = (an+ 2/an)/2 and a1 = 1.
    This gives ( 1 , 3/2 , 17/12 , 577/408 , 665857/470832 , ... ) which is approximately ( 1, 1.5, 1.41667, 1.414215, 1,414213562, ... )

    This is the result of applying Newton's method for solving an equation to x2 = 2 and hence gives a method of calculating sqrt2.

  4. an+1 = an + 1/n2 and a1 = 0.
    This gives the sequence ( 0 , 1 , 5/4 , 49/36 , 205/144 , 5269/3600 , ... ) which is approximately ( 0, 1, 1.25, 1.3611, 1.423611, 1.463611, ... )
    These are the partial sums of the series sigmai1inf(1/i2) which the Swiss mathematician Leonhard Euler (1707 to 1783) proved "settles down" to 1.6449... = p2/6.

One may think of a sequence as being used to approximate a real number which might be difficult to get hold of directly. For example, the sequence 6) above gives approximations to the number sqrt2.

The main thing to remember is:

Informal definition

A real sequence ( an) is said to be convergent to a limit alpha if all the terms of the sequence become close (*) to alpha for n large (**).

To formalise the idea at (*), we mean that if we are given any small error epsilon then the terms of the sequence are within epsilon of alpha provided n is big enough. By big enough (**) we mean that we can find some N so that this happens when n > N.

We get the rigorous statement corresponding to the above:

Definition

A real sequence ( an) converges to a limit alpha if:
given epsilon > 0, there exists N belongs N such that if n > N then | an- alpha | < epsilon.

Remarks

  1. The N that it is necessary to choose will depend on what epsilon you are using. In general, the smaller the epsilon the bigger you will have to choose N.
  2. Note that all the terms an with n > N must approximate alpha to better than epsilon.
  3. In terms of quantifiers we may express this as:
    A sequence (xi) rarrow alpha if (forallepsilon > 0) (thereexistsN belongs N)(forall n > N)(|xn - alpha| < epsilon)
    Note that since the "(thereexistsN belongs N)" occurs after the "(forallepsilon > 0)" the value of N that we must find is allowed to depend upon epsilon.
  4. A sequence which does not converge is called divergent.

Examples

  1. The sequence ( 1 , 1/2 , 1/3 , ... ) is convergent to 0.

    Proof
    Given epsilon use the Archimedian property to choose N with 1/Nlte epsilon. Then if n > N we have 1/n < 1/N lte epsilon.


  2. The sequence ( 1, 2, 3, 4, ... ) is not convergent (to any limit).

    Proof
    We will see later than any unbounded sequence does not converge.



  3. The sequence (1, 3/2 , 11/6 , 25/12 , 137/60 , 49/20 , ... ) = (1 , 1 +1/2 , 1 +1/2+1/3 , 1 +1/2+1/3+1/4 , ... ), where an= sigmai1n(1/i), is not convergent.

    Proof
    The terms of this sequence are the partial sums of the harmonic series sigmai1inf(1/i). This result was first proved by Jacob Bernoulli (1654 to 1705)
    Group the terms of the series in the following way.
    1 + 1/2 + (1/3+1/4) + (1/5+1/6+1/7+1/8) + (1/9+1/10+ ... +1/16) + ...
    and these terms are bigger than the terms of
    1 + 1/2+ (1/4+1/4) + (1/8+1/8+1/8+1/8) + (1/16+1/16+ ... +1/16) + ... = 1 + 1/2 + 1/2 + 1/2 + ...
    and the partial sums of this series are clearly unbounded.


  4. The sequence ( 1, 0, 1, 0, 1, 0, 1, 0, ... ) is not convergent even though it is bounded.

Previous page
(Axioms for the Real numbers) 		Contents Next page
(Some properties of convergent sequences)
JOC September 2002