 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/15/d/solution

Factor out

x^{2}

before expanding the logarithm:

lo g (1 + x + x^{2}) = 2 lo g x + lo g (1 + \frac{1}{x} + \frac{1}{x ^{2}})

for large positive

x

. Using

lo g (1 + u) = u - u^{2} /2 + O (u^{3})

then gives

lo g (1 + x + x^{2}) = 2 lo g x + \frac{1}{x} + \frac{1}{2 x ^{2}} + O (x^{- 3}), x \to + \infty .

These are the first three nonzero terms, including the leading logarithm. Replacing

lo g x

lo g ∣ x ∣

yields the corresponding two-sided large-

∣ x ∣

expansion.

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