if
n is a non negative integer, we can use Pascal's Triangle or the
formula formula for the binomial expansion,
A Level Maths Notes: C4 – The General Binomial Theorem
To expand the expressionif
n is a non negative integer, we can use Pascal's Triangle or the
formula formula for the binomial expansion,
We can only use the formula
above if
is
a non negative integer. Ifis
negative or a fraction we can still expand the expression, but must
use the formula for the general binomial expansion:
The expansion is infinite, and in practice only a finite number of terms are found.
Example: Expand
up
to and including the term in
Simplifying the constant for each term gives
For each general binomial
expansion of the type give above,
must
be restricted to a certain interval, called the interval of
convergence. For the the example above
It
is no coincidence that
is
then positive, since we are taking the square root of this expression
in this question.
Example: Find the binomial
expansion of
up
to and including the term in
Simplifying the constant for each term gives
The above expansion is only
valid for