What is the coefficient of x^3 in the Maclaurin series for ln(1+x)?

What is the coefficient of x^3 in the Maclaurin series for ln(1+x)?

• A. -1/3
• B. 0
• C. 1/3
• D. 1/2

Answer: (C)

The coefficient of x^3 comes from the Maclaurin series for ln(1+x), which is found by integrating the geometric series for 1/(1+x). For |x|<1, 1/(1+x) = 1 - x + x^2 - x^3 + … . Integrating term by term gives ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + … . The x^3 term is positive and has coefficient 1/3, since the general term is (-1)^{n+1} x^n / n and n=3 yields (+1)/3.

The coefficient of x^3 comes from the Maclaurin series for ln(1+x), which is found by integrating the geometric series for 1/(1+x). For |x|<1, 1/(1+x) = 1 - x + x^2 - x^3 + … . Integrating term by term gives ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + … . The x^3 term is positive and has coefficient 1/3, since the general term is (-1)^{n+1} x^n / n and n=3 yields (+1)/3.