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

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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...

...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...

...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...

...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...

...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,...

...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...

...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...

...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...