MT2002 Analysis

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## Convergence in the Reals

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

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(πn/4)) = (1/√2 ,1 , 1/√2 , 0 , -1/√2 ,-1 , -1/√2 , 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 √2.

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 (1/i2) which the Swiss mathematician Leonhard Euler (1707 to 1783) proved "settles down" to 1.6449... = π2/6.

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 √2.

The main thing to remember is:

Informal definition

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

To formalise the idea at (*), we mean that if we are given any small error ε then the terms of the sequence are within ε of α 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 α if:
given ε > 0, there exists NN such that if n > N then | an- α | < ε.

Remarks

1. The N that it is necessary to choose will depend on what ε you are using. In general, the smaller the ε the bigger you will have to choose N.
2. Note that all the terms an with n > N must approximate α to better than ε.
3. In terms of quantifiers we may express this as:
A sequence (xi)→ α if ( ε > 0) ( NN)( n > N)(|xn - α| < ε)
Note that since the "( NN)" occurs after the "( ε > 0)" the value of N that we must find is allowed to depend upon ε.
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 ε use the Archimedian property to choose N with 1/Nε. Then if n > N we have 1/n < 1/Nε. 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= (1/i), is not convergent.

Proof
The terms of this sequence are the partial sums of the harmonic series (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.

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JOC September 2001