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- One of the two most important ideas in Real analysis is that of
- A
**sequence**in**R**is a*list*or*ordered set*: (*a*_{1},*a*_{2},*a*_{3}, ... ) of real numbers. **Examples**One may define a sequence (

*a*_{n}) by giving an explicit formula for the*n*th term.

- (
^{1}/_{n}) = ( 1 ,^{1}/_{2},^{1}/_{3}, ... ) - (sin(
^{πn}/_{4})) = (^{1}/_{√2},1 ,^{1}/_{√2}, 0 , -^{1}/_{√2},-1 , -^{1}/_{√2}, 0 , ... ) - (
^{(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*n*th term in terms of earlier ones.*a*_{n+1}=*a*_{n}/(*a*_{n}+ 1),*a*_{1}= 1.

This gives the sequence ( 1 ,^{1}/_{2},^{1}/_{3},^{1}/_{4}, ... ).*f*_{n+2}=*f*_{n+1}+*f*_{n}and*f*_{1}=*f*_{2}= 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).

*a*_{n+1}= (*a*_{n}+ 2/*a*_{n})/2 and*a*_{1}= 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

*x*^{2}= 2 and hence gives a method of calculating √2.*a*_{n+1}=*a*_{n}+ 1/*n*^{2}and*a*_{1}= 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... = π^{2}/_{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 (
*a*_{n}) 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 (
*a*_{n})**converges to a limit***α*if:

given*ε*> 0, there exists*N*∈**N**such that if*n*>*N*then |*a*_{n}-*α*| <*ε*. **Remarks**- 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*.

- Note that
*all*the terms*a*_{n}with*n*>*N*must approximate*α*to better than*ε*.

- In terms of quantifiers we may express this as:

A sequence (*x*_{i})→*α*if (*ε*> 0) (*N*∈**N**)(*n*>*N*)(|*x*_{n}-*α*| <*ε*)

Note that since the "(*N*∈**N**)" occurs after the "(*ε*> 0)" the value of*N*that we must find is allowed to depend upon*ε*.

- A sequence which does not converge is called
**divergent**.

**Examples***The sequence*( 1 ,^{1}/_{2},^{1}/_{3}, ... )*is convergent to*0.**Proof**

Given*ε*use the Archimedian property to choose*N*with^{1}/_{N}≤*ε*. Then if*n*>*N*we have^{1}/_{n}<^{1}/_{N}≤*ε*.

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

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

*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**a*_{n}= (^{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.

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

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- The

**Definition**