A Level Further Mathematics Core Pure Practice Test

The Maclaurin expansion for ln(1+x) around 0 up to x^3 starts with which term?

x - x^2/2 + x^3/3 + ... x + x^2/2 + x^3/3 + ... x - x^2/2 - x^3/3 + ... x + x^2/2 - x^3/3 + ...

Correct Answer:

x - x^2/2 + x^3/3 + ...

Explanation:

Start with the pattern of the Maclaurin expansion for ln(1+x). It comes from integrating the geometric series for 1/(1+x), which is 1 - x + x^2 - x^3 + ... 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 chosen so ln(1+0) = 0. So up to x^3, the expansion is x - x^2/2 + x^3/3. The x^2 term must be negative and the x^3 term positive, which is consistent with this pattern and rules out the other sign choices.

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