## Approximation in series expansion

This question was posed by a student.

Explanation

Exact versus approximation. The two curves depart from each other at x=pi/5

When we use the approximation to compute the volume, it differs significantly from the exact volume.

## Challenging Maclaurin’s series problem 1

If ,

(i) find the Maclaurin’s series of f(x) up to and including the term in .

(ii) Hence deduce the series expansion of the function  up to and including the term in

Solution

(i) is routine, can easily work out to be

(ii) requires Higher Order Thinking (HOT)

We differentiate the series obtained in (i) to get

Principle used: Differentiate one series expansion to get the series expansion for another series.

## Quick Maclaurin’s series expansion using GC

In Maclaurin’s series expansion, where up to f”(0) need to be evaluated, can be done quickly using the GC.

Example

Expand  up to powers of 2.

Using TI 84 Plus (OS 2.55)

From G.C,

f(0)= 3/2

f'(0) = -3/4

f”(0)= 5/4

Therefore using Maclaurin’s series expansion,

If powers higher than 2 such as 3 are required, then do differentiation to find the first derivative. Then enter into Y1. Y2 and Y3 will then give the second and third derivative.