 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/11/b/ii/solution

Near zero the logarithm's argument is positive, so

ln [(2 + x)^{3}] = 3 ln (2 + x) = 3 ln 2 + 3 ln (1 + x /2)

ln [(2 + x)^{3}] = 3 ln 2 + \frac{3}{2} x - \frac{3}{8} x^{2} + \frac{1}{8} x^{3} + O (x^{4}) .

x > - 2

for the real logarithm; the power series converges for

∣ x ∣ < 2

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