Welcome to the World of Infinite Series: Binomial Expansion (P4)
Hello mathematicians! Welcome to a fascinating chapter where we take the familiar Binomial Expansion and give it a powerful upgrade. In P1/P2, you learned how to expand expressions like \((x+y)^4\). But what if the power was negative, or even a fraction, like \((1+x)^{-2}\) or \(\sqrt{1+x}\)?
In Unit P4, we tackle this challenge using the Binomial Series. This tool is incredibly important because it allows us to turn complicated roots and reciprocal functions into simple polynomials, which is essential for advanced calculus and approximations. Don't worry if this seems tricky at first—we'll break it down step-by-step!
Quick Review: The Familiar Binomial (Prerequisite Knowledge)
Before diving into P4 territory, remember the basic form of the expansion you learned previously, which only works when the power \(n\) is a positive integer:
When \(n\) is a positive integer, the expansion stops after \(n+1\) terms. We use the notation \(\binom{n}{r}\) for the coefficients.
Example: \((1+x)^3 = 1 + 3x + 3x^2 + x^3\). It stops!
The P4 Binomial Series: Negative and Fractional Powers
When the power \(n\) is negative or a fraction (a non-positive integer rational number), the expansion never stops. It becomes an infinite series.
The General Formula for \((1+x)^n\)
The P4 curriculum requires you to use the standard formula for the expansion of \((1+x)^n\), provided that \(|x| < 1\).
The formula is:
\[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots \]
Breaking Down the Terms (Coefficients)
- Term 1: Always \(1\).
- Term 2: \(nx\)
- Term 3: The coefficient is \(\frac{n(n-1)}{2}\). (Remember that \(2! = 2 \times 1 = 2\)).
- Term 4: The coefficient is \(\frac{n(n-1)(n-2)}{6}\). (Since \(3! = 3 \times 2 \times 1 = 6\)).
Notice the pattern: The denominator is the factorial of the power of \(x\), and the numerator has exactly that many factors starting with \(n\).
Step-by-Step Example: Expanding \((1+x)^{-2}\)
Here, \(n = -2\). We usually need to find the first four terms (up to \(x^3\)).
- Term 1 (Constant): \(1\)
- Term 2 (Coefficient of \(x\)): \(nx = (-2)x = -2x\)
- Term 3 (Coefficient of \(x^2\)): \[ \frac{n(n-1)}{2!}x^2 = \frac{(-2)(-2-1)}{2}x^2 = \frac{(-2)(-3)}{2}x^2 = \frac{6}{2}x^2 = 3x^2 \]
- Term 4 (Coefficient of \(x^3\)): \[ \frac{n(n-1)(n-2)}{3!}x^3 = \frac{(-2)(-3)(-2-2)}{6}x^3 = \frac{(-2)(-3)(-4)}{6}x^3 = \frac{-24}{6}x^3 = -4x^3 \]
Therefore, \((1+x)^{-2} \approx 1 - 2x + 3x^2 - 4x^3 + \dots\)
When \(n\) is a negative integer (like -1, -2, -3...), the signs of the terms in the expansion often alternate: \(+ - + - \dots\) This is a great way to quickly check your work!
Key Takeaway: For non-positive integer powers, we use an infinite series formula where the coefficients are derived by multiplying \(n\) by descending integers.
The Critical Condition for Validity: \(|x| < 1\)
This is arguably the most common oversight in P4 Binomial questions! Since the expansion is an infinite series, it will only give a correct, finite value if the terms eventually get smaller and smaller. This concept is called convergence.
Why is convergence necessary?
Think of it like throwing a bouncing ball. If the ball bounces lower each time, it eventually stops (converges). If the power \(n\) is negative or fractional, we need the terms \(nx\), \(x^2\), \(x^3\), etc., to diminish quickly.
This happens only when the value being raised to the power (which is \(x\) in the standard formula \((1+x)^n\)) is small. Specifically, its magnitude must be less than 1.
The Condition:
\[ |x| < 1 \quad \text{or} \quad -1 < x < 1 \]
Students often expand correctly but forget to state the validity range. Always check what the expansion is valid for!
Did you know? If \(|x| \ge 1\), the terms of the series would grow larger or stay the same, meaning the series would run away to infinity (it diverges), giving an incorrect answer.
Key Takeaway: The P4 Binomial Series is only valid when the variable term has a magnitude strictly less than 1.
Handling Complex Forms: \((a+bx)^n\)
The standard P4 formula only works for \((1+\mathbf{x})^n\). If you are asked to expand something more general like \((4+x)^{1/2}\) or \((8-3x)^{-1}\), you must algebraically manipulate it first.
The Mandatory Factorisation Step
You must factor out the first constant term, \(a\), so that the first term inside the bracket becomes \(1\).
\[ (a+bx)^n = \left[ a \left( 1 + \frac{b}{a}x \right) \right]^n = a^n \left( 1 + \frac{b}{a}x \right)^n \]
Step-by-Step Example: Expanding \((4-x)^{-1/2}\)
Here, \(n = -1/2\), \(a = 4\), and \(b = -1\).
Step 1: Factor out \(a\).
\[ (4-x)^{-1/2} = \left[ 4 \left( 1 - \frac{1}{4}x \right) \right]^{-1/2} \]
Step 2: Apply the power \(n\) to the constant factor.
\[ = 4^{-1/2} \left( 1 - \frac{1}{4}x \right)^{-1/2} \]
Since \(4^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}\):
\[
= \frac{1}{2} \left( 1 - \frac{1}{4}x \right)^{-1/2}
\]
Step 3: Define the new variable \(X\).
Let \(X = -\frac{1}{4}x\). We now expand \((1+X)^n\), where \(n = -1/2\).
The expansion of \((1+X)^{-1/2}\) starts: \[ 1 + nX + \frac{n(n-1)}{2!}X^2 + \dots \] \[ 1 + \left(-\frac{1}{2}\right)\left(-\frac{1}{4}x\right) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2}\left(-\frac{1}{4}x\right)^2 + \dots \] \[ 1 + \frac{1}{8}x + \frac{3/4}{2}\left(\frac{1}{16}x^2\right) + \dots \] \[ 1 + \frac{1}{8}x + \frac{3}{32} \left(\frac{1}{16}x^2\right) + \dots = 1 + \frac{1}{8}x + \frac{3}{512}x^2 + \dots \]
Step 4: Multiply by the initial constant factor (\(1/2\)).
\[ (4-x)^{-1/2} = \frac{1}{2} \left( 1 + \frac{1}{8}x + \frac{3}{512}x^2 + \dots \right) \] \[ = \frac{1}{2} + \frac{1}{16}x + \frac{3}{1024}x^2 + \dots \]
Determining the Validity Range for \((a+bx)^n\)
The expansion is valid when the term substituted for \(x\) in the standard formula is between -1 and 1.
In the example above, our substituted term was \(-\frac{1}{4}x\).
We require: \[ \left| -\frac{1}{4}x \right| < 1 \] \[ \frac{1}{4}|x| < 1 \] \[ |x| < 4 \quad \text{or} \quad -4 < x < 4 \]
1. Convert: Always rewrite the expression into the form \(A(1 + X)^n\).
2. Expand: Use the formula on \((1+X)^n\).
3. Multiply: Multiply the result by the factor \(A\).
4. Range: Determine the range based on \(|X| < 1\).
Key Takeaway: Never apply the formula directly to \((a+bx)^n\). Always factor out the constant \(a\) first, and ensure you use this factor in your final answer and range calculation.
Applications of the Binomial Expansion: Approximations
One of the most powerful uses of the P4 Binomial Series is to estimate the values of roots or reciprocals without needing a calculator. This is done by substituting a small value into the derived series.
Process for Approximations
Goal: Use the expansion of \((4-x)^{-1/2}\) to approximate \(1/\sqrt{3.9}\).
Step 1: Relate the expression to the required approximation.
We want to approximate \(1/\sqrt{3.9}\). Note that \(1/\sqrt{3.9} = (3.9)^{-1/2}\).
We need \((4-x)^{-1/2} = (3.9)^{-1/2}\).
Therefore, \(4-x = 3.9\), which means \(x = 0.1\).
Step 2: Check Validity.
We found the expansion is valid for \(|x| < 4\). Since \(x=0.1\) is well within this range, the approximation will be accurate.
Step 3: Substitute \(x\) into the expansion.
Using the expansion we found earlier: \[ (4-x)^{-1/2} \approx \frac{1}{2} + \frac{1}{16}x + \frac{3}{1024}x^2 \] Substitute \(x = 0.1\): \[ (3.9)^{-1/2} \approx \frac{1}{2} + \frac{1}{16}(0.1) + \frac{3}{1024}(0.1)^2 \] \[ \approx 0.5 + 0.00625 + 3 \times (0.00009765625) \] \[ \approx 0.5 + 0.00625 + 0.00029296875 \] \[ \approx 0.50654296875 \]
(The actual value of \(1/\sqrt{3.9}\) is approximately 0.50637, showing the approximation is very close!)
This method works because when \(x\) is very small, the terms involving \(x^2\), \(x^3\), etc., become tiny, and the expansion quickly converges to the correct value.
Key Concepts and Final Tips
Summary of Essential Skills
- Identify the power \(n\) and the variable term \(X\).
- Use the standard Binomial Series formula for fractional or negative \(n\).
- Factor out the constant term \(a\) from \((a+bx)^n\) to get the form \(A(1+X)^n\).
- Simplify coefficients involving fractions and negative numbers carefully.
- Always state the condition for validity, \(|X| < 1\), and solve for \(|x|\).
Watch Out for Signs!
When \(n\) is negative or fractional, it's very easy to make sign errors, especially in the numerator of the coefficients.
Example: If \(n = -1/2\):
- \(n-1 = -1/2 - 1 = -3/2\)
- \(n-2 = -1/2 - 2 = -5/2\)
Be meticulous when multiplying these negative fractions together!
A Note on Factorials
While factorials (\(r!\)) are used in the denominators of the general formula, when dealing with specific numerical terms (like \(n=1/2\)), it is often easier just to divide by the required number (2 for \(x^2\), 6 for \(x^3\), etc.) rather than using the full \(\binom{n}{r}\) notation, which strictly applies only to positive integers.
You've got this! Mastering the Binomial Series opens doors to complex mathematical modeling and is a cornerstone of advanced Pure Mathematics. Keep practicing that factorisation step and checking those validity ranges!