 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/15/d/solution

Combine the logarithms first and factor out the limiting constant:

ln (1 + x + 2 x^{2}) - ln x^{2} = ln (2 + \frac{1}{x} + \frac{1}{x ^{2}}) = ln 2 + ln (1 + \frac{1}{2 x} + \frac{1}{2 x ^{2}}) .

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

with

u = 1/ (2 x) + 1/ (2 x^{2})

. The first three nonzero terms, including the constant, are

ln (1 + x + 2 x^{2}) - ln x^{2} = ln 2 + \frac{1}{2 x} + \frac{3}{8 x ^{2}} + O (x^{- 3}), x \to \infty.

This is an asymptotic expansion in reciprocal powers of

x

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