Title: HELM 2020 - questions being used.
Course: HELM Workbooks 2023-2024

Maclaurin and Taylor series

As we shall see, many functions can be represented by power series. In fact we have already seen in earlier Sections examples of such a representation: \[\begin{aligned} \dfrac{1}{1-x}&=1+x+x^2+\cdots\qquad |x| <1\\ \ln (1+x)&=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots \qquad -1 < x \leq 1\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ e^x&=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots\qquad \text{all}\ x\end{aligned}\] The first two examples show that, as long as we constrain \(x\) to lie within the domain \(|x|<1\) (or, equivalently, \(-1<x<1\)), then in the first case \(\dfrac{1}{1-x}\) has the same numerical value as \(1+x+x^2+\cdots\) and in the second case \(\ln (1+x)\) has the same numerical value as \(x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots\).

In the third example we see that \(e^x\) has the same numerical value as \(1+x+\dfrac{x^2}{2!}+\cdots\) but in this case there is no restriction to be placed on the value of \(x\) since this power series converges for all values of \(x\). The figure below shows this situation geometrically. As more and more terms are used from the series \(1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}\cdots\) the curve representing \(e^x\) is a better and better approximation. In (a) we show the linear approximation to \(e^x\). In (b) and (c) we show, respectively, the quadratic and cubic approximations.

[Image: Linear, quadratic and cubic approximations to e^x]
Linear, quadratic and cubic approximations to \(e^x\)

These power series representations are extremely important, from many points of view. Numerically, we can simply replace the function \(\dfrac{1}{1-x}\) by the quadratic expression \(1+x+x^2\) as long as \(x\) is so small that powers of \(x\) greater than or equal to 3 can be ignored in comparison to quadratic terms. This approach can be used to approximate more complicated functions in terms of simpler polynomials. Our aim now is to see how these power series expansions are obtained.

Where to find the theory

See section 9.7.4 of Engineering Mathematics, or 8.7.4 of Mathematics for the Natural Sciences text for the necessary theory needed to complete these exercises.

Each question asks you to optimised (find the extreme points) of a given functions given some constrains. In solving this we construct an auxiliary function often called the Lagrangian. If we want to optimised \(f(x,y)=0\) subject the the constrain \( g(x,y)\) then we introduce the function
\[ f(x,y)-lg(x,y)\]
involving a new variable \(l\). The solution to the problem is given by the solutions to system of equations coming from the (three) first order partial derivatives of the new function. Please see the book for more details before attempting these exercises.