Introduction:

The nth degree Taylor polynomial is one of the interesting topics in mathematics. nth degree Taylor polynomials are the most elementary functions. Because nth degree Taylor polynomials involve only the basic arithmetic operations. Differentiate and long-term behavior is in nth degree Taylor polynomial. It is very easy to understand. Here we discuss about the nth degree Taylor polynomial and example problems involved in nth degree taylor polynomial.

 

Definition for finding nth degree taylor polynomial:

 

An approximation of a function with nth degree polynomial can be defined as the function of f.  It can be differentiate for n times. The nth degree Taylor polynomial for centered at a is

Pn(x) = f(a) + f′(a)(x - a) + (f″/ 2!)(x – a) ²+ …… + (f (n) (a)/ n!)(x - a)n

= ∑n k = 0 (f (k) (a) / k!) (x- a)k

 

Properties on nth degree taylor polynomial:

 

  • If the function is analytic than the series of converges for x in the interval (a + r, a - r) and the result is equal to f(x).
  • The power series representation can be done in algebraic expression. It can be used in the simplest form of Euler’s formula.
  • Harmonic analysis is one of the fundamentals of fields are the result of Taylor series expansions of cosine, sine and exponential functions.
  • Taylor functions are sometimes cannot write as function that is singularity.
  • The function is not analytic then the Taylor series is zero. Also the function is not.

 

Example problem on finding nth degree taylor polynomial:

 

Example problem:

Solve: f(x) = 1 / 1- x, a = 0. Find the nth degree Taylor polynomial.

Solution:

Given f(x) = 1 / 1- x

f’(x) = 1 / (1 – x) ²     f’ (0) = 1

f”(x) =  2 / (1 – x)²    f’’(0) = 2

f’’’(x) = 6 / (1 – x) ²   f’’’ (0) = 6

To find the nth degree Taylor polynomial

Pn(x) = f(a) + f′(a)(x - a) + (f″/ 2!)(x – a) ²+ …… + (f (n) (a)/ n!)(x - a) n

= f(0) + f’(0) (x – a) + f’’(0)((x – a)²/ 2!) + f’’’ (0) ((x – a) ³/ 3!) +……..

= 1 + 2((x – 0)² / 2!) + 6 (x – 0)³/ 6 + …..

= 1 + 2x² + x³+ …