Extending the Binomial Theorem for fractional powers
Newton’s work on the Binomial Theorem
The use of the Binomial Theorem was known to mathematicians before Newton, but Newton
generalised it during his exile from Cambridge, during the ‘plague’ years. Around 1665 he found how
to expand the binomial into a series for fractional powers. He realised this is possible as he was
attempting to calculate the areas of curvilinear surfaces. He noted that, since area under
y  x n for the interval [0, x] is equal to
x n1
n 1
[1]
that he can use this further to find the result for the area under the curve itself by assuming that the
area of a curve can be represented as a sum of the infinite series of areas of each term.
Remember;
The binomial expansion formula
What happens when n is a fraction? Simply replace n by the fraction.
Example;

Consider the expansion (1 + x)1/2;
1  x 
1
2
1 1
1  1  3 
 

    
1
2  2  2 2  2  2  3
 1 x 
x 
x  ...
2
2!
3!
1
1
1
 1  x  x 2  x3  ...
2
8
16
Expansion is valid for x  1  1  x  1
n
1/4
n(n-1)
n(n-1)(n-2)
(1 + x)n
-1/2
3/5
Challenge; Can you do this?
Find an approximation for 1/√1+x