Question

Give the definition of sequence of real numbers (Xn). Then give 2 example.

Answer 1

First I give you an intuitive idea of sequence of real numbers. You can think it as an infinite list of real numbers, so you have that two such infinite lists are equal if and only if their first numbers coincide, their second numbers coincide, their third numbers coincide, and so on. Observe that if we change the position of the numbers we could obtain another infinite list.

This is the precise definition: A sequence of real numbers is a function
`X:\mathbb{N}\to \mathbb{R}`. Where
`\mathbb{N}` and
`\mathbb{R}` are natural and real numbers respectively.

Note that I called the function
`X` for convenience; for the next reason. Usually we write
`X_n` instead of
`X(n)` for the value at
`n` of the function
`X`.

So if we think the function
`X` as a list, we have that its first number is
`X_1=X(1)` (or
`X_0=X(0)` if we admite
`0\in \mathbb{N}`), its second number is
`X_2=X(2)`, and so on.

For the examples you can give an algebraicaly manner to obtain the terms of the sequence (the values of the function
`X`). So for example the sequence
`X_1=1 , X_2=4, X_3=9, \dots` is given by the formula

`X_n=X(n) = n^2`

Or for example a more complicated sequence given by de formula
`X(n)=X_n=\frac {n+1}{2n^4+2}`.