Calculus course

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Calculus course: Sequences and Series of functions

A sum is the result of an addition. The notion of series is related to the sum of numbers.

For example, adding 1, 2, 3, and 4 gives the sum 10: 1+2+3+4=10. The decad represents the number ten.

“Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: If one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1 + 2 + 3 + 4 = 10). And if one exceeds the tetrad, one will exceed 10 too…. So that the number by the unit resides in the number 10, but potentially in the number 4.” (Aetius 1.3.8)

The Tetractys (greek Τετρακτύς) is a symbol composed of ten dots in an upward-pointing triangular formation.

The numbers being summed are called addends.

Given a set A, a sequence of  elements of A is a function F: N -> A; rather than using notation F(n) for the elements that have been selected from A, since the domain is always the natural numbers (N), we use the notation a_n=F(n) and denote the sequence:

{a_n} or a_1,a_2,a_3...

Given any sequence {c_n} of elements  of a set A, we have an associated sequence of nth partial sums:

{s_n} where s_n= \sum{c_k} with k=1…n

the symbol \sum{c_k} is called series (or infinite series).

We take A to be the set of real functions on R.

Pointwise convergence of sequence of functions.

Definition: let {f_n} be a sequence of functions defined on a set of real numbers A. We say that {f_n} converges pointwise to a function f on A for each x \in A, the sequence of real numbers {f_n(x)} converges to the number f(x). In other words, for each x \in A, we have:


Example: let f_n(x)=x^n, x \in [0,1] and let f(x)= 0 if 0 <=x<1, f(x)=1 if x=1.

Then {f_n(x)} converges to f pointwise on [0,1]

Suppose {f_n(x)} converges to f pointwise on A. Then given \epsilon >0, and given x \in A , there exists  N=N(x, \epsilon) \in I such that:

|f_n(x)-f(x)|< \epsilon for all n>=N

In general N depends on \epsilon a.w.a x.

Example: f_n(x)= \frac{x}{1+nx}.

We have f_n(x) <=1/n for all x \in [0, oo( and hence  for a given \epsilon >0, any N  with N>\frac{1}{\epsilon} will imply that |f_n(x)-0|< \epsilon for all  n>N and for x \in [0, oo(

Uniform  convergence of sequence of functions

Let { f_n(x)} be a sequence of function on A. We say that { f_n(x)} converges uniformly  to f on A if for given \epsilon >0, there exists  N=N(\epsilon) , depending on \epsilon , such that:

|f_n(x)-f(x)|< \epsilon for all n>=N and for all x \in A

If { f_n(x)} converges  uniformly to f on A, we write:  f_n -> f uniformly on A.

Remark: N depends only on \epsilon and not on x.

Example: f_n(x)= \frac{x}{1+nx} converges uniformly to 0 on [0, oo(

If f_n -> 0 uniformly on A and \epsilon >0 , then there exists  N>0 and all x \in A , |f_n(x)|< \epsilon. This implies  that for all n>N:

sup|f_n(x)|<= \epsilon and hence lim sup|f_n(x)|<= \epsilon.

Since \epsilon>0 is an arbitrary positive number, we conclude that:

if  f_n(x) ->0 uniformly on A, then lim_{x\to\infty} sup|f_n(x)|=0.

The converse is also true.


if  f_n(x) ->f uniformly on A, if and only if  lim_{x\to\infty} sup|f_n(x)-f(x)|=0.

Cauchy Criterion for Uniform Convergence. A sequence {f_n(x)} converges uniformly on A if and only if for a given \epsilon>0 , there exists N>0 such that for all n>=m>N and for all x\in A,

$latex| f_n(x)-f_m(x)|< \epsilon $

Theorem: If  {f_n(x)} is a sequence  of continuos functions on a bounded  and closed interval $ latex [a,b] and  {f_n(x)} converges pointwise to a continuous function f on [a,b], then f_n -> f uniformly on [a,b]

Consequences of uniform convergence

Theorem: if f_n -> f uniformly on [a,b], if f_n are continuous at c \in [a,b] , then f is continuous at c.

Corollary: if f_n -> f uniformly on [a,b],if f_n are continuous on [a,b], then f is continuous on [a,b].

Supermath collection: Calculus Course

Calculus is a branch of mathematics that deals with rates of change. Its roots go back as far as Ancient Greece, but calculus as we know it today began with Newton and Leibnitz, in the 17th century. Basic ideas of calculus include the idea of limit , derivative , and integral. In the sciences, many processes involving change( or related variables) are studied. Calculus is a powerful tool to study the ways in which the variables interact.

RATIONAL AND IRRATIONAL NUMBERSNumbers are classified according to type. The first type of number is the first type you ever learned about: the counting, or “natural” numbers (N):

1, 2, 3, 4, 5 and so on

If we include 0 (zero), we have the “whole” numbers (W):

0, 1, 2, 3, 4, 5 …etc…etc…

Then come the “integers” (Z), if we include their algebraic negatives (zero, the natural numbers, and the negatives of the naturals):

… –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5 …

But the necessary numbers are the rationals and irrationals.
Real numbers are either rational or irrational. The word “rational” (Q)comes from the word “ratio.” A number is rational if it can be expressed as the quotient, or ratio, of two whole numbers. Any whole number is rational. Its denominator is 1.

Any rational number can be expressed as the quotient of two integers in many ways. For example,


-> As a fraction a
, where a and b are integers (b 0).

An irrational number is a real number that cannot be expressed as the ratio of two integers. For example:
\sqrt2= 3,14...

A real number has a decimal representation.  It gives the approximate location of the number on the real line.

Most commonly, “real line” is used to mean real axis, a line with a fixed scale so that every real number corresponds to a unique point on the line. The term “real line” is also used to distinguish an ordinary line from a so-called imaginary line.

Any real number except 0 (zero) is either negative or positive. Numbers to the left of zero on the number line are negative numbers. They are written with a minus sign (e.g., -2).
Numbers to the right of 0 on the number line are positive numbers. They may be written with a plus sign (e.g., +2), but usually the plus sign is omitted. Thus, real numbers may be referred to as signed numbers.

+C_0,C_1,C_2...C_n.... or -C_0,C_1,C_2...C_n...


A function is a relation that uniquely associates members of one set with members of another set.  More formally, a function from A to B is an object f such that every a \in A is uniquely associated with an object f(b) \in B. A function is therefore a many-to-one relation. The domain of a function is the complete set A of possible values of the independent variable in the function. The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain. Domain and range can be seen clearly from a graph.

For example: consider the function f(x)= \frac{1}{x-2}.

Notice that there is no output number when the input number equals 2, since division by zero is meaningless. Thus x can have any value except 2. We say that 2 is not in the domain of the function f.

Here is another example. Let f(x)= \sqrt(x-4)

Since the quantity under the radical must be non-negative x-4>=0. Thus the implicit domain is [4,+oo[

Another one: let f(x)= \frac{1}{x\sqrt(x-1)}.

Since we cannot divide by 0, we know that x cannot equal to either 0 or 1.  And since we cannot take the square root of a negative number (remember, output numbers must be real), then x-1>=0. Taken together, x-1>=0 and x \neq0, these requirements mean that x-1>0. Thus the implicit domain is (1, +oo)

Example: Find the range of function f defined by: f(x)= x^2-2

The domain of this function is the set of all real numbers. The range is the set of values that f(x) takes as x varies. If x is a real number, x2 is either positive or zero. Hence we can write the following: x^2>=0. Subtract -2 to both sides to obtain x^2-2>=-2. The last inequality indicates that x^2- 2 takes all values greater that or equal to -2. The range of f is given by [-2, +oo( . A graph of f also helps in interpreting the range of a function. Below is shown the graph of function f given above. Note the lowest point in the graph has a y (= f (x) ) value of -2.

Example: f(x)= \sqrt (x+4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y >=0.