 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/9/b/solution

Adding and subtracting the exponential function power series cancels the appropriate odd or even terms:

sinh x = n = 0 \sum \infty \frac{x ^{2 n + 1}}{( 2 n + 1 )!} = x + \frac{x ^{3}}{3 !} + \dots, cosh x = n = 0 \sum \infty \frac{x ^{2 n}}{( 2 n )!} = 1 + \frac{x ^{2}}{2 !} + \dots .

Both series converge absolutely for every complex number

x

, as follows from the ratio test.

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