 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/20/a/i/solution

Let

a_{n} = 2^{n} / n!

. The ratio test gives

a_{n + 1} / a_{n} = 2/ (n + 1) \to 0 < 1

, so the series converges absolutely. In fact the exponential series identifies its value as

\sum_{n = 1}^{\infty} \frac{2 ^{n}}{n !} = e^{2} - 1,

with the

n = 0

term subtracted.

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