Q. 0 - Calculus

Question

Read the section and make your own summary of the material.

Step-by-Step Solution

Answer

The summary of the material is that the only difference between the Taylor series and the Maclaurin series is that Using zero as our single point, a Maclaurin polynomial is a particular instance of the Taylor polynomial.

Step 1: Definition of Maclaurin series and Taylor series

• Any terms in a Maclaurin series expansion must be a non-negative integer power of the variable. 
• A function is represented as an infinite sum of terms determined from the values of its derivatives at a single point by the Taylor Series, also known as the Taylor Polynomial.
• The difference between the Taylor series and the Maclaurin series is that Using zero as our single point, a Maclaurin polynomial is a particular instance of the Taylor polynomial.

Step 2: Summary of the section

A power series is an infinite series involving the positive powers of a variable x.

       f(x) = $a_o$ + $a_{1}x$ + $a_2{x}^2$ + · · · = $\sum _{n=0}^{n=\infty }{a}_n{x}^n$ 

The radius of convergence R of the power series is a real number in the range of [0,$\infty$).

A polynomial function is a power series in which the sum converges for all x.

Types of power series:

1) Taylor series:

Taylor series formula is given as

                                     $\sum _{n=0}^{\infty }\frac{f^n\left(0\right)}{n!}{x}^n$

when |x| < R where R is the radius of convergence of the power series above.

2) Maclaurin series:

The difference between the Taylor series and the Maclaurin series is that Using zero as our single point, a Maclaurin polynomial is a particular instance of the Taylor polynomial.