What are the first three nonzero terms of the Maclaurin series for ln(1+x)?

What are the first three nonzero terms of the Maclaurin series for ln(1+x)?

• A. x + x^2/2 - x^3/3
• B. x - x^2/2 - x^3/3
• C. x - x^2/2 + x^3/3
• D. x + x^2/2 + x^3/3

Answer: (C)

This question uses the Maclaurin expansion of ln(1+x) obtained by integrating the geometric series for 1/(1+x) around 0. The series for 1/(1+x) is 1 - x + x^2 - x^3 + x^4 - ..., valid for |x| < 1. Integrating term by term from 0 to x gives ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ..., with the constant determined by ln(1+0) = 0. The signs alternate, and the n-th term is (-1)^{n+1} x^n / n. Hence the first three nonzero terms are x - x^2/2 + x^3/3.

This question uses the Maclaurin expansion of ln(1+x) obtained by integrating the geometric series for 1/(1+x) around 0. The series for 1/(1+x) is 1 - x + x^2 - x^3 + x^4 - ..., valid for |x| < 1. Integrating term by term from 0 to x gives ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ..., with the constant determined by ln(1+0) = 0. The signs alternate, and the n-th term is (-1)^{n+1} x^n / n. Hence the first three nonzero terms are x - x^2/2 + x^3/3.