Taylors Theorem:
Taylor's theorem gives an approximation of a n times differentiable function around a given point by a n-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are fixed order truncations of its Taylor's series, which completely determines the function in some locality of the point. There are numerous forms of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor-polynomial.

Some examples of Taylor’s theorem are:

Ex. 2) Expand log tanπ4+x in ascending orders of x.

Example 3: In order to write or calculate a Taylor series for we first need to calculate its n -derivatives, which we have already done above. The Taylor Series is defined as:

Simplifying it we get:

The easiest number to choose for a is probably 1, though you can choose whatever number you want to for a , so long as its n derivatives are all defined at a. Substituting a for 1 gives:

Now let us evaluate f(x) at x=6 using the Taylor Series:

Relationship to Analyticity.
Taylor’s expansion of real analytical functions
Let I⊂R be an open interval. By definition, a function f:I→R is real analytic if it is locally defined by a convergent power series. This means that for every a ∈ Ithere exists some r > 0 and a sequence of coefficients ck ∈ R such that (a − r, a + r) ⊂ I and

In general, the radius of convergence of a power series can be computed from the Cauchy–Hadamard formula

This result is based on comparison with a geometric series, and the same method shows that if the power series based on a converges for some b∈R, it must converge uniformly on the closed interval [a − rb, a + rb], where rb = |b − a|. Here only the convergence of the power series is considered, and it might well be that(a − R,a + R) extends beyond the domain I of f. The Taylor polynomials of the real analytic function f at a are simply the finite truncations

...Historical Account:
Pythagoras, the namesake and supposed discoverer of the Pythagorean Theorem, was born on the Greek island of Samos in the early in the late 6th century. Not much is known about his early years of life, however, we do know that Pythagoras traveled through Egypt in the attempt to learn more about mathematics.
Besides his famous theorem, Pythagoras gained fame for founding a group, the Brotherhood of Pythagoreans, which was dedicated solely to study of mathematics and worship of numbers. Pythagoras passed on his belied that numbers are in fact the true "rulers of the universe".
While studying in Egypt, it is believed that Pythagoras studied with people known as the "rope-stretchers", the same people who engineered the pyramids. By using a special form of a rope tied in a circle with 12 evenly spaced knots, they discovered that if the rope was pegged to the ground in the dimensions of 3-4-5, the rope would create a right triangle. The rope stretchers used this principle to help accurately lay the foundations of for their buildings.
It was this fascination with the rope stretchers 3-4-5 triangle that ultimately led to the discovery of the Pythagorean theorem. While experimenting with this concept by drawing in the sand, Pythagoras found that if a square is drawn from each side of the 3-4-5 triangle, the area of the two smaller squares could be added together and equal the area of the large square....

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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including...

...BINOMIAL THEOREM :
AKSHAY MISHRA
XI A , K V 2 , GWALIOR
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set.
HISTORY :
HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century...

...The Coase Theorem
In “The Problem of Social Cost,” Ronald Coase introduced a different way of thinking about externalities, private property rights and government intervention. The student will briefly discuss how the Coase Theorem, as it would later become known, provides an alternative to government regulation and provision of services and the importance of private property in his theorem.
In his book The Economics of Welfare, Arthur C. Pigou, a British economist, asserted that the existence of externalities, which are benefits conferred or costs imposed on others that are not taken into account by the person taking the action (innocent bystander?), is sufficient justification for government intervention. He advocated subsidies for activities that created positive externalities and, when negative externalities existed, he advocated a tax on such activities to discourage them. (The Concise, n.d.). He asserted that when negative externalities are present, which indicated a divergence between private cost and social cost, the government had a role to tax and/or regulate activities that caused the externality to align the private cost with the social cost (Djerdingen, 2003, p. 2). He advocated that government regulation can enhance efficiency because it can correct imperfections, called “market failures” (McTeer, n.d.).
In contrast, Ronald Coase challenged the idea that the government had a role in taking action targeted...

...The Sylow theoremsThe Sylow Theorems
Here is my version of the proof of the Sylow theorems. It is the result of
taking the proof in Gallian and trying to make it as digestible as possible. In
particular, I tried to break the long proof into bite-sized pieces. The main
goal here is to convey an overview of how the ingredients fit together, so I'll
skip lightly over some of the details.
The prerequisites are basically all of the group theory that came before the
Sylow theorems in this course, including: Lagrange's theorem, the first and
second isomorphism theorems, and the orbit-stabilizer theorem. I'll also use
Cauchy's theorem, even though the book lists it as a corollary to the Sylow
theorems (more on that later). I'll assume you know the definition of a
Sylow-subgroup and all the terms in the statements of the Sylow theorems.
From now on, G is a finite group and p is a prime number that divides the order
of G. Recall that a p-subgroup of G is a subgroup of G with order equal to a
power of p.
Definition. A maximal p-subgroup of G is a p-subgroup of G that is not contained
in any larger p-subgroup of G.
This is only a temporary definition, since it will turn out that a "maximal
p-subgroup" is just the same thing as a "Sylow p-subgroup". However it would be
circular logic to assume this. Hypothetically, you could...

...Negative Externalities and the Coase Theorem
As Adam Smith explained, selfishness leads markets to produce whatever people want. To get rich, you have to sell what the public wants to buy. Voluntary exchange will only take place if both parties perceive that they are better off. Positive externalities result in beneficial outcomes for others, whereas negative externalities impose costs on others. The Coase Theorem is most easily explained via an example
This paper addresses a classic example of a negative externality (pollution), and describes three possible solutions for the problem: government regulation, taxation and property right – a better solution to overcome the externality as described by economist Ronald Coase.
Imagine being a corn farmer and growing corn. What are the private costs that you face that help you determine production? Things like fuel, seed, fertilizer; these are your private costs. But it turns out that every spring and summer when you lay down the fertilizer some of this flows into the stream nearby and flows into a lake downstream, oftentimes resulting in large fish kills. All those downstream, the fisherman, the recreationist, and the landowners all incur a negative externality.
There are three ways in which we can address these externalities:
1- Government Regulation:
a) First, direct regulation is applied through technology-specific methods. This is where the government requires producers to use a...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that...

...Thevenin Theorem
It provides a mathematical technique for replacing for a given network, as viewed from two output terminals by a single voltage source with a series resistance. It makes the solution of complicated networks (particularly, electronic networks) quite quick and easy.
The Thevenin’s theorem, as applied to d.c. circuits, may be stated as under:
The current flowing through a load resistance RL connected across any two terminals A and B of a linear, active bilateral network is given by VOC || (Ri + RL) where VOC is the open-circuit voltage (i.e. voltage across the two terminals when RL is removed) and Ri is the internal resistance of the network as viewed back into the open-circuited network from terminals A and B with all voltage sources replaced by their internal resistance (if any) and current sources by infinite resistance.
How to Thevenize a given circuit?
1. Temporarily remove the resistance (called load resistance RL) whose current is required.
2. Find an open-circuit voltage VOC which appears across the two terminals from where resistance has been removed. It is also called the Thevenin voltage Vth.
3. Compute the resistance of the whose network as looked into from these two terminals after all voltage sources have been removed by living behind their internal resistances (if any) and current sources have been replaced by open-circuit i.e. infinite resistance. It is also called Thevenin resistance Rth or Ri.
4. Replace...