Why the graph of a sequence cannot be a line or a curve?

Answer 1

A sequence can be thought of a list of numbers written in a definite order:

a1, a2, a3, ..., a(n+1), an, ...

The number a1 is called the first term, a2 is the second term, and in general an is the nth term.

For every positive integer n, there is a corresponding number an, and so a sequence can be defined as a function whose domain is the set of all positive integers. But, we usually write an instead of the function f(n) for the value of the function at the number n.

Notation: The sequence {a1, a2, a3, ...} is also denoted by {an}.

Some sequences can be defined by giving a formula for the nth term. For example:

an = (n)/(n +1) the sequence is {1/2, 2/3, 3/4, 4/5, ..., (n)/(n+1), ...}

this sequence can be pictured either by plotting its terms on a number line, or by plotting its graph. Since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinates (1, a1), (2, a2), (3, a3), ..., (n, an).

Try to plot the graph of this sequence, and you will see that the terms of the sequence an = n/(n+1) are approaching 1 as n becomes large. In this case 1 is the limit for this sequence, and we can write:

lim n -->∞ an = 1