(1 + x)n, where is rational can be written as a series:
Unless n is a positive integer, this is an infinite series.
This series will converge if |n| < 1.
Example : Expand up to the term in x3.
Using the expansion of (1 + x)n and substituting -2x for x and n =
Note: this series is only valid for |-2x| < 1 ie valid for |x| <
The binomial expansion can be used to find approximations to irrational roots.
Find to 2dp an approximation to .
If we let x = 0.01 then
From the series expansion above
= 10 (0.989994995+..)
= 9.90 to 2dp.