The Binomial Expansion
Neil Trivedi
Teacher
Binomial Expansion
The Binomial Expansion is used to expand expressions by the form , where is any real number.
In Year 12, we focused on cases where was a natural number. The expansion is given by the formula:
where the binomial coefficient is
Example 1:
Using the formula for , expand .
Single Step: Apply the formula to find each term, then sum them together.
Sum the terms together.
No answer provided.
When is negative or a fraction, our calculators cannot compute binomial coefficients. In these cases, we use an alternative formula.
This formula can still be used when is a natural number, notice how continuing to subtract will eventually lead to , meaning the expansion ends at some point. When is negative or a fraction, subtracting leads to an infinite series. For that reason, exams will specify exactly how many terms they want.
Example 2:
Use the binomial expansion to find the first terms of in ascending powers of .
Step 1: Rewrite the expression on one line.
Step 2: Expand using the binomial formula, up to and including the term.
No answer provided.
In the next example, the expression contains a constant that is not . We factor it out so that the term inside the bracket has as its constant, allowing us to apply the binomial expansion formula.
Example 3:
Find the series expansion of in ascending powers of up to and including the term.
Step 1: Rewrite the expression to a suitable form by factoring out . Remember to maintain the power on what you have factorised out as per our index rule .
Step 2: Expand using the formula, up to and including the term.
No answer provided.
Validity of Expansions
For a binomial expansion to be valid, it must converge rather than diverge. When is a natural number, the expansion is finite and therefore is always valid. When is negative or a fraction, the expansion is infinite, so we must determine the values of for which it converges.
The series converges when and diverges when . Therefore, it's valid for .
In simpler English, the values of that you substitute into your expansion must lead to the infinite sum to approach a finite value, also known as the limit. This happens when the size of is between and .
In general, when is not a natural number, the expansion of is valid when .
This can be shown by factoring out , giving
For example, , which can be rewritten as , is valid when
Example 4:
For the previous expansions of and , what would the series expansions be valid for?
Single Step: We want to ensure that the expansion produces a series that converges.
So, .
First, let's consider the validity for .
Now, let’s consider the validity for .
No answer provided.
Example 5:
a) Find the series expansion of in ascending powers of up to and including the term.
Step 1: Rewrite the expression to a suitable form for expansion. Do this by factoring out .
Step 2: Expand using the formula, up to and including the term.
b) State the values of for which it is valid.
Single Step: We want to ensure that the expansion produces a series that converges.
So, .
c) Use this expansion to find an approximation for .
Step 1: Find the value of to substitute into the expansion. We equate to and solve for .
Step 2: Substitute the value of , found in step 1, into the expansion to find the approximation.
No answer provided.
Challenging Question