Question

Match List-I with List-II and choose the correct option:

\[ \begin{array}{|l|l|} \hline \textbf{LIST-I (Function)} & \textbf{LIST-II (Expansion)} \\ \hline A. \log(1-x) & I. 1 + \frac{1}{3} + \frac{1}{6} + \frac{3}{40} + \frac{15}{336} + \dots \\ \hline B. \sin^{-1} x & II. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \\ \hline C. \log 2 & III. x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \dots, -1 < x \le 1 \\ \hline D. \frac{\pi}{2} & IV. -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots, -1 \le x < 1 \\ \hline \end{array} \]

• A. A-IV, B-III, C-II, D-I
• B. A-III, B-IV, C-I, D-II
• C. A-III, B-IV, C-II, D-I
• D. A-I, B-II, C-III, D-IV

Hint:

Memorize the basic Maclaurin series for functions like \( e^x, \sin x, \cos x, \log(1+x), (1+x)^p \). Many other series, like \( \sin^{-1} x \), can be derived from these by differentiation or integration. For matching questions, even if one pair seems obscure, finding the other correct pairs can often lead you to the right answer.

Answer 1

The Correct Option is A

Objective: Match functions/constants to their Taylor/Maclaurin series. Procedure: A. \( \log(1-x) \): The Maclaurin series for \( \log(1+u) \) is \( u - \frac{u^2}{2} + \frac{u^3}{3} - ... \). With \( u = -x \), the series becomes: \[ \log(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - ... \] This corresponds to IV. B. \( \sin^{-1} x \): The series for \( \sin^{-1} x \) is derived by integrating the series for \( (1-x^2)^{-1/2} \). The expansion is: \[ \sin^{-1} x = x + \frac{1}{2}\frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4}\frac{x^5}{5} + ... \] This corresponds to III. C. \( \log 2 \): This value is obtained from the series for \( \log(1+x) \) at \( x=1 \). The series is: \[ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... \] Setting \( x=1 \) yields the alternating harmonic series: \[ \log 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... \] This corresponds to II. D. \( \frac{\pi}{2} \): Given the previous definitive matches, \( \frac{\pi}{2} \) must correspond to series I. While the provided series I differs from the standard expansion of \( \sin^{-1} x \) at \( x=1 \) (which equals \( \frac{\pi}{2} \)), the established pairings for A, B, and C confirm the overall correct matching. Conclusion: The confirmed pairings are A-IV, B-III, C-II, and D-I, aligning with option (A).