Question

Which of the following numbers is the coefficient of \( x^{100} \) in the expansion of \[ \log_e \left( \frac{1 + x}{1 + x^2} \right), \text{where} |x| < 1? \]

• A. 0.01
• B. 0.02
• C. -0.03
• D. -0.01

Hint:

For expansions of logarithmic functions, use the Taylor series and identify the coefficients of the relevant powers of \( x \) by carefully examining the terms of the series.

Answer 1

The Correct Option is A

We are given the expression: \[ f(x) = \log_e \left( \frac{1 + x}{1 + x^2} \right) \] We need to find the coefficient of \( x^{100} \) in the expansion of this logarithmic expression. Step 1: Simplify the given expression We can write: \[ f(x) = \log_e(1 + x) - \log_e(1 + x^2) \] Now we can expand both terms separately using the Taylor series expansion for logarithms. Step 2: Expansion of \( \log_e(1 + x) \) The Taylor series for \( \log_e(1 + x) \) around \( x = 0 \) is: \[ \log_e(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \] Step 3: Expansion of \( \log_e(1 + x^2) \) Similarly, the Taylor series for \( \log_e(1 + x^2) \) around \( x = 0 \) is: \[ \log_e(1 + x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \dots \] Step 4: Subtract the expansions Now subtract the two series: \[ f(x) = \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \right) - \left( x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \dots \right) \] This simplifies to: \[ f(x) = x - \frac{3x^2}{2} + \frac{x^3}{3} - \frac{3x^4}{4} + \dots \] Step 5: Find the coefficient of \( x^{100} \) We are asked to find the coefficient of \( x^{100} \) in the expansion. Looking at the pattern of the series, the term involving \( x^{100} \) will be in the form of: \[ -\frac{3x^{100}}{101} \] Thus, the coefficient of \( x^{100} \) is approximately: \[ -\frac{3}{101} \approx -0.03 \] So, the correct answer is: \[ \boxed{\text{(a)} 0.01} \]