UCL Online STEP and AEA Preparation Sessions
Session 10: The binomial expansion and approximations
The binomial expansion provides you with a quick way to expand brackets like
 a  b  a  b  a  b  a  b  a  b  a  b 
.
It also provides you with series with finitely many terms which approximate functions
1
like f ( x) 
.
3
2
1  2 x 
Computers use series such as those provided by the binomial expansion to evaluate
certain functions. This lecture will take a look at how the binomial expansion can be
used to produce approximations and how judgements about their accuracy can be
made.
Try evaluating the following using your calculator
1
1  1.5 1012
.
The answer is not exactly 1. The binomial expansion will provide insight into this.
This session will also cover the general manipulation of series produced using the
binomial expansion and deductions that can be made from such series.
Example problem
Use the binomial expansion of
number which approximates
9  x as far as the term in x with x 
23 .
1
to find a rational
64