Taylor Series

Wednesday, May 29, 2013

Taylor Series

Taylor series is a infinit number of terms.Taylor series is a expansion of series and its function about a point.The Taylor series of a real or composite function ƒ(x) that is much differentiable in a neighborhood of a real or composite number a is the power series

that can be written in the more compact sigma notation as

where n! indicates the factorial of n and ƒ⁽ⁿ⁾(a) indicates the nth derivative of ƒ calculate at the point a.

I like to share this Taylor Series Expansion Example with you all through my article.

Explanation of taylor series:

Maclaurin Series:
If a=0, then it is said to be maclaurin series.

Derivation:
we define the power series as,

At x=0,

Differentiate the function,

At x = 0,

Differentiating again will give,

At x=0, we will evaluate the equation as,

Generalizing the equation,we get

Substitute the values of a_n in the power expansion,

Generalizing f in a more general form, we have

Evaluating at x = a, we get

Sustitute above equation, we get taylor series.

        Taylor series can be used to evaluate the value of an whole function in each point, if the functional value and its derivatives are identified at single point. Uses of the Taylor series for whole functions are:
      1.The sum of partial series can be used as approximations of the entire function.
        2.The representation of series reduces many mathematical proofs.
       In Taylor series, algebraic functions are indicated using an algebraic equation, and transcendental functions are indicated using properties which holds them, namely differential equation. Example the exponential function is equal to its own derivative and its original value is 1.
       Taylor series are used to identify functions and operators in diverse areas of mathematics. Example: Analytical functions of matrices and operators can be defined as matrix exponential. In formal analysis, we directly work with the power series .