 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-b/3/a/ii/solution

The expression in the PDF is

1/ a^{x} - 1

, with the subtraction outside the fraction. Using the exponential series with

a^{- x} = e^{- x l n a}

gives

a^{- x} - 1 = - (ln a) x + \frac{( ln a ) ^{2}}{2} x^{2} + O (x^{3}) .

For

a > 0

a \neq = 1

, these are the first two nonzero terms. If

a = 1

, the function is identically zero, so there are no nonvanishing terms.

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