1 The binomial series

A very important infinite series which occurs often in applications and in algebra has the form:

$1+px+\frac{p\left(p-1\right)}{2!}{x}^2+\frac{p\left(p-1\right)\left(p-2\right)}{3!}{x}^3+\cdots$

in which $p$ is a given number and $x$ is a variable. By using the ratio test it can be shown that this series converges, irrespective of the value of $p$ , as long as $\left|x\right|<1$ . In fact, as we shall see in Section 16.5 the given series converges to the value ${\left(1+x\right)}^p$ as long as $\left|x\right|<1$ .

The binomial theorem can be obtained directly from the binomial series if $p$ is chosen to be a positive integer (here we need not demand that $\left|x\right|<1$ as the series is now finite and so is always convergent irrespective of the value of $x$ ). For example, with $p=2$ we obtain

With $p=3$ we get

Generally if $p=n$ (a positive integer) then

${\left(1+x\right)}^n=1+nx+\frac{n\left(n-1\right)}{2!}{x}^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{x}^3+\cdots +x^n$

which is a form of the binomial theorem. If $x$ is replaced by $\frac{b}{a}$ then

${\left(1+\frac{b}{a}\right)}^n=1+n\left(\frac{b}{a}\right)+\frac{n\left(n-1\right)}{2!}{\left(\frac{b}{a}\right)}^2+\cdots +{\left(\frac{b}{a}\right)}^n$

Now multiplying both sides by $a^n$ we have the following Key Point:

Task!

Use the binomial theorem to obtain

1. ${\left(1+x\right)}^7$
2. ${\left(a+b\right)}^4$

1. Here $n=7$ :

   Answer

   ${\left(1+x\right)}^7=1+7x+21x^2+35x^3+35x^4+21x^5+7x^6+x^7$

2. Here $n=4$ :

   Answer

   ${\left(a+b\right)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4.$

Task!

Given that $x$ is so small that powers of $x^3$ and above may be ignored in comparison to lower order terms, find a quadratic approximation of ${\left(1-x\right)}^{\frac{1}{2}}$ and check for accuracy your approximation for $x=0.1$ .

First expand ${\left(1-x\right)}^{\frac{1}{2}}$ using the binomial series with $p=\frac{1}{2}$ and with $x$ replaced by $\left(-x\right)$ :

Answer

Now obtain the quadratic approximation:

Answer

Now check on the validity of the approximation by choosing $x=0.1$ :

Answer

What we have done in this last Task is to replace (or approximate) the function ${\left(1-x\right)}^{\frac{1}{2}}$ by the simpler (polynomial) function $1-\frac{1}{2}x-\frac{1}{8}{x}^2$ which is reasonable provided $x$ is very small. This approximation is well illustrated geometrically by drawing the curves $y={\left(1-x\right)}^{\frac{1}{2}}$ and $y=1-\frac{1}{2}x-\frac{1}{8}{x}^2$ . The two curves coincide when $x$ is ‘small’. See Figure 2:

Figure 2

Task!

Obtain a cubic approximation of $\frac{1}{\left(2+x\right)}$ . Check your approximation for accuracy using appropriate values of $x$ .

First write the term $\frac{1}{\left(2+x\right)}$ in a form suitable for the binomial series (refer to Key Point 9):

Answer

Now expand using the binomial series with $p=-1$ and $\frac{x}{2}$ instead of $x$ , to include terms up to $x^3$ :

Answer

State the range of $x$ for which the binomial series of ${\left(1+\frac{x}{2}\right)}^{-1}$ is valid:

Answer

Choose $x=0.1$ to check the accuracy of your approximation:

Answer

Figure 3 below illustrates the close correspondence (when $x$ is ‘small’) between the curves $y=\frac{1}{2}{\left(1+\frac{x}{2}\right)}^{-1}$ and $y=\frac{1}{2}-\frac{x}{4}+\frac{x^2}{8}-\frac{x^3}{16}$ .

Figure 3

Exercises

1. Determine the expansion of each of the following
   1. ${\left(a+b\right)}^3$ ,
   2. ${\left(1-x\right)}^5$ ,
   3. ${\left(1+x^2\right)}^{-1}$ ,
   4. ${\left(1-x\right)}^{1∕3}$ .
2. Obtain a cubic approximation (valid if $x$ is small) of the function ${\left(1+2x\right)}^{3∕2}$ .

Answer