4.2. SEQUENCES 55

Polynomials and rational functions are particularly easy functions to understand be-cause they do come from a simple formula.

Definition 4.1.10 A function f given by f (x) = anxn + an−1xn−1 + · · ·+ a1x+ a0is called a polynomial. Here the ai are real or complex numbers and n is a nonnegativeinteger. In this case the degree of the polynomial f (x) is n. Thus the degree of a polynomialis the largest exponent appearing on the variable.

f is a rational function if f (x) = h(x)g(x) where h and g are polynomials.

For example, f (x) = 3x5 +9x2 +7x+5 is a polynomial of degree 5 and 3x5+9x2+7x+5x4+3x+x+1

is a rational function.Note that in the case of a rational function, the domain of the function might not be all

of F. For example, if f (x) = x2+8x+1 , the domain of f would be all complex numbers not

equal to −1.Closely related to the definition of a function is the concept of the graph of a function.

Definition 4.1.11 Given two sets, X and Y, the Cartesian product of the two sets,written as X×Y, is assumed to be a set described as follows.

X×Y = {(x,y) : x ∈ X and y ∈ Y} .

F2 denotes the Cartesian product of F with F. Recall F could be either R or C.

The notion of Cartesian product is just an abstraction of the concept of identifying apoint in the plane with an ordered pair of numbers.

Definition 4.1.12 Let f : D( f )→ R( f ) be a function. The graph of f consists ofthe set,

{(x,y) : y = f (x) for x ∈ D( f )} .

Note that knowledge of the graph of a function is equivalent to knowledge of the func-tion. To find f (x) , simply observe the ordered pair which has x as its first element and thevalue of y equals f (x) .

4.2 SequencesFunctions defined on the set of integers larger than a given integer are called sequences.

Definition 4.2.1 A function whose domain is defined as a set of the form

{k,k+1,k+2, · · ·}

for k an integer is known as a sequence. Thus you can consider

f (k) , f (k+1) , f (k+2) ,

etc. Usually the domain of the sequence is either N, the natural numbers consisting of{1,2,3, · · ·} or the nonnegative integers, {0,1,2,3, · · ·} . Also, it is traditional to writef1, f2, etc. instead of f (1) , f (2) , f (3) etc. when referring to sequences. In the abovecontext, fk is called the first term, fk+1 the second and so forth. It is also common to writethe sequence, not as f but as { fi}∞

i=k or just { fi} for short.