 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/15/b/ii/solution

h = x + 1

, the displacement from the expansion point. The numerator has Taylor series

sinh h = h + h^{3} /6 + O (h^{5})

, while the reciprocal denominator has geometric series

(1 + h)^{- 1} = 1 - h + h^{2} - h^{3} + h^{4} + O (h^{5})

\frac{sinh ( x + 1 )}{x + 2} = (x + 1) - (x + 1)^{2} + \frac{7}{6} (x + 1)^{3} - \frac{7}{6} (x + 1)^{4} + O ((x + 1)^{5}) .

The local power series converges for

∣ x + 1∣ < 1

, the distance to the denominator's zero at

x = - 2

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