Question 3 of 4beginner🔧 ApplyLong Answer4 marks

Verify that (1+x)−2=1−2x+3x2−4x3+…(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots (when ∣x∣<1|x| < 1).

Correct Answer

To verify (1+x)−2=1−2x+3x2−4x3+…(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots (when ∣x∣<1|x| < 1):

We use the Binomial expansion for (1+x)m(1+x)^m when mm is a negative integer. The expansion is given by:

(1+x)m=1+mx+m(m−1)1⋅2x2+m(m−1)(m−2)1⋅2⋅3x3+…(1+x)^m = 1 + mx + \frac{m(m-1)}{1 \cdot 2}x^2 + \frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}x^3 + \ldots

Here, m=−2m = -2. Substituting m=−2m = -2 in the expansion:

First term: 11

Second term: mx=(−2)x=−2xmx = (-2)x = -2x

Third term:

m(m−1)1⋅2x2=(−2)(−2−1)2x2=(−2)(−3)2x2=62x2=3x2\frac{m(m-1)}{1 \cdot 2}x^2 = \frac{(-2)(-2-1)}{2}x^2 = \frac{(-2)(-3)}{2}x^2 = \frac{6}{2}x^2 = 3x^2

Fourth term:

m(m−1)(m−2)1⋅2⋅3x3=(−2)(−3)(−4)6x3=−246x3=−4x3\frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}x^3 = \frac{(-2)(-3)(-4)}{6}x^3 = \frac{-24}{6}x^3 = -4x^3

Therefore, (1+x)−2=1−2x+3x2−4x3+…(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \ldots

Note: The condition ∣x∣<1|x| < 1 is necessary when mm is a negative integer, as stated in the textbook. Without this condition, the series would not converge.

Exercise: A.1.2 Binomial Theorem for any Index | Q: 3 | (Page Page 2)
For More Understanding

Explanation

The question requires verifying the binomial expansion of (1+x)−2(1+x)^{-2}. The textbook context provides the general formula for binomial expansion when mm is negative, along with the important remark that ∣x∣<1|x| < 1 is a necessary condition for convergence. The solution applies the general term formula systematically to derive each coefficient.

Solution Steps

  1. Step 1: Identify m=−2m = -2 and apply the binomial expansion formula for negative integer powers

  2. Step 2: Calculate the first term as 11

  3. Step 3: Calculate the second term as mx=−2xmx = -2x

  4. Step 4: Calculate the third term using m(m−1)2=62=3\frac{m(m-1)}{2} = \frac{6}{2} = 3, giving 3x23x^2

  5. Step 5: Calculate the fourth term using m(m−1)(m−2)6=−246=−4\frac{m(m-1)(m-2)}{6} = \frac{-24}{6} = -4, giving −4x3-4x^3

  6. Step 6: Conclude the expansion matches the given series and note the convergence condition ∣x∣<1|x| < 1

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