Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Consider the geometric series \(1+x+x^{2}+x^{3}+…\)
a) For which values of \(x\) does this series converge?
b) When the series converges, what is the value of the sum to infinity of the series?
Solutions
Get Ready
Questions
From the Required Prior Knowledge we know that$$\frac{1}{1-x}=\left(1-x\right)^{-1}=1+x+x^{2}+x^{3}+…$$when \(\left|x\right|<1, x\ne 0\).
Use this to write \(\left(1-x\right)^{-2}\) as an infinite sum.
HINT - we know that \(\left(1-x\right)^{-2}=\left(\left(1-x\right)^{-1}\right)^{2}=\left(1-x\right)^{-1}\left(1-x\right)^{-1}\)
In a similar way, write \(\left(1-x\right)^{-3}\) as an infinite sum.
Solutions
Notes
We can generalise this result to get the binomial expansion of$$\left(1+x\right)^{n}$$where \(n\in\mathbb{Q}\) and \(-1<x<1, x\ne 0\).$$\left(1+x\right)^{n}=1+nx+\frac{n\left(n-1\right)}{2!}x^{2}+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^{3}+…$$
Examples and Your Turns
Example
Expand \(\sqrt{1+2x}\) for \(\left|x\right|<\frac{1}{2}\), up to the term in \(x^{3}\).
Your Turn
Expand \(\frac{2}{1-3x}\) for \(\left|x\right|<\frac{1}{3}\), up to the term in \(x^{3}\).
Notes
That result is only useful if the binomial is of the form \(1+b\) for some \(b\in\mathbb{Q}\).
Fortunately we can take any binomial of the form \(\left(a+b\right)^{n}\) and factorise it into this simpler form:$$\begin{align}\left(a+b\right)^{n}&=\frac{a^{n}\left(a+b\right)^{n}}{a^{n}}\\&=a^{n}\frac{\left(a+b\right)^{n}}{a^{n}}\\&=a^{n}\left(\frac{\left(a+b\right)}{a}\right)^{n}\\&=a^{n}\left(1+\frac{b}{a}\right)^{n}\end{align}$$Now this is of the form above so we get the following formula, which is given in the formula booklet:$$\left(a+b\right)^{n}=a^{n}\left(1+n\left(\frac{b}{a}\right)+\frac{n\left(n-1\right)}{2!}\left(\frac{b}{a}\right)^{2}+…\right)$$It is worth recalling that this only worked when \(-1<x<1, x\ne 0\) as it was derived from the infinite sum of a geometric series (see the Required Prior Knowledge above).
So this formula only works when \(\left|\frac{b}{a}\right|<1\). This is not stated in the formula booklet.
Examples and Your Turns
Example
Expand \(\frac{1}{\sqrt{2+x}}\) up to the term in \(x^{3}\).
State the interval of convergence for the expansion.
Your Turn
Expand \(\frac{1}{\sqrt{3+2x}}\) up to the term in \(x^{3}\).
State the interval of convergence for the expansion.
Your Turn
Use the binomial theorem to show that$$\sqrt{\frac{1+x}{1-x}}\approx 1+x+\frac{1}{2}x^{2}$$when \(\left|x\right|<1\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.