**Taylor
Series** is the expansion of the function.
It represents function as infinite sum of the derivative values of the
function. The series was introduced by Brook Taylor. The series is
approximation of a real-complex function around a point. This series if
developed around zero is referred to as **Maclaurin Series**, introduced by **Colin
Maclaurin**, a Scottish mathematician.

**What is Taylor Series?**

The **Taylor Series** is
polynomial approximation of a function. It is nothing but the "**infinite
degree**" of Taylor polynomial. It is a special kind of power series.

Let's say, there is a function which is infinitely differential at; then the series centered around will be

The one dimensional series for the
same function can be written as

Suppose a= 0, then the series
becomes Maclaurin series.

**Examples of Taylor Series**

Find the first four non zero terms of the Taylor series centered at for

Also, this can regarded as taylor
polynomial of Tanx for degree 7.

What is the Taylor Series?

To find the series, we have start with
the derivates of

sin’(x) = cos (x)

sin”(x) = - sin(x)

sin””(x) = - cos (x)

sin^{4}(x) = sin(x)

Now, we will find the function and
derivates

sin(0) = 0

sin’(0) = 1

sin”(0) = 0

sin””(0) = -1

By Taylors formula,

**How Taylor Series Work?**

Taylor Series follows Taylors
theorem which states that if

and it hold the condition

Then the series is,

The proof of the theorem is based on
continuous application of the L'Hôpital's rule
such that

The Taylor series is basically a
power series. It is summation of series of differential taken around a
particular point. Lets understand how it works with help of an example

Suppose a function needs to be
approximate around a point .Let's assume point to be zero. As the function is
continuous let take a constant function y = f(0). then,

Y’ = f’(0)

Y = f’(0)x + c

As x = 0, f(0), approximation will
be

y = f’(0) x + f(0)y

The second derivative would be

y" = f”(0)

y’ = f”(0)x + f’(0)

y = f”(0) x^{2}/2 + f’(0)x +
f(0)

The third derivative would be

y = f”(0)x36 + f”(0) x^{2}/2
+ f’(0)x + f(0)

The approximation will addition of
all these, it will be

**Application of Taylor Series**

The
foremost important application of the series is the calculation of the function
to the approximate value. In algorithm series are used indirectly, for instance
algorithm CORDIC uses series to find the value of arctan2^{-n }before
hard coding them. It is also used in Numerical analysis such as error
approximation and error bound estimation.

The
series expansion has helped in the development of many other advanced
techniques that manipulates using past approximation for cancelling the error.
one such example is Romberg procedure for integration.

It have some practical application in physic such as it is used to develop pendulum equation and Einstein Brownian motion. Taylor Series calculator is also widely used in applied mathematics and engineering. Although the series is being widely used in various subject it have certain constraints such as it is polynomial and just approximation. It shows error if moved away from the point of expansion.

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