A sequence is just a list of numbers. Here are a few examples:
1,2,3,4,5 , ...
1, 2, 4, 6, 8, 10, 12, ...
2, 4, 8, 16, 32, 64, 128,...
The elements of the sequence need not be integers:
1, 1/2, 1/4, 1/8, 1/16, ...
is a sequence whose limit (as i -> infty) is zero. A general sequence is written out as
a1 , a2 , a3, a4 ,...., ai , ...
its ith element is the number ai , pronounced ``ay-sub-eye''.
Alternatively, a sequence is a map a: N -> R where N denotes the counting numbers {1,2,3, ...} and R denotes the real numbers. The two notations for sequences are linked by a(i) = a i .
Sometimes we can find explicit formulae for the elements of a sequence. In the examples above we have that the sequences can be expressed by the formulae:
a i = i
a i = 2i
a i = 2i
and
a i = 1/2i
The basic question we will usually ask about a sequence is ¿ does it converge? Of the sequences above,only the last one converges to a finite number, namely to zero: = 1/(2i -> 0 as i -> : read ``one over two to the i tends to zero as i tends to infinity''. All the other sequences converge to + infinity as i- > infinity.
Remarks. As a happy coincidence, all the sequences just described can be obtained by taking standard real valued functions on the real number line R and restricting them to the counting numbers N which is a subset of R . These functions are, respectively
f(x) = x
f(x) =2x
f(x) = 2 x
and
f(x) = 2 -x .