Binomial Theorem: Fractional & Negative Indices
How do I use the binomial theorem for fractional and negative indices?
- The formula given in the formula booklet for the binomial theorem applies to positive integers only
-
- where
- For negative or fractional powers the expression in the brackets must first be changed such that the value for a is 1
- This is given in the formula booklet
- If a = 1 and b = x the binomial theorem is simplified to
- This is not in the formula booklet, you must remember it or be able to derive it from the formula given
- You need to be able to recognise a negative or fractional power
- The expression may be on the denominator of a fraction
- Or written as a surd
- The expression may be on the denominator of a fraction
- For the expansion is infinitely long
- You will usually be asked to find the first three terms
- The expansion is only valid for
- This means
- This is known as the interval of convergence
- For an expansion the interval of convergence would be
How do we use the binomial theorem to estimate a value?
- The binomial expansion can be used to form an approximation for a value raised to a power
- Since higher powers of x will be very small
- Usually only the first three or four terms are needed to form an approximation
- The more terms used the closer the approximation is to the true value
- The following steps may help you use the binomial expansion to approximate a value
- STEP 1: Compare the value you are approximating to the expression being expanded
- e.g.
- STEP 2: Find the value of x by solving the appropriate equation
- e.g.
- STEP 1: Compare the value you are approximating to the expression being expanded
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- STEP 3: Substitute this value of x into the expansion to find the approximation
- e.g.
- STEP 3: Substitute this value of x into the expansion to find the approximation
- Check that the value of x is within the interval of convergence for the expression
- If x is outside the interval of convergence then the approximation may not be valid
Exam Tip
- Students often struggle with the extension of the binomial theorem questions in the exam, however the formula is given in the formula booklet
- Make sure you can locate the formula easily and practice substituting values in
- Mistakes are often made with negative numbers or by forgetting to use brackets properly
- Writing one term per line can help with both of these
Worked Example
Consider the binomial expansion of .
a)
Write down the first three terms.
b)
State the interval of convergence for the complete expansion.
c)
Use the terms found in part (a) to estimate . Give your answer as a fraction.