 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/9/b/solution

L = lim_{n \to \infty} ∣ a_{n + 1} / a_{n} ∣

, when it exists for eventually nonzero terms:

L < 1

L > 1

gives divergence, and

L = 1

is inconclusive.

For

a_{n} = n / (2^{n} - 1)

\frac{a _{n + 1}}{a _{n}} = \frac{n + 1}{n} \frac{2 ^{n} - 1}{2 ^{n + 1} - 1} ⟶ \frac{1}{2} .

Thus the first series converges absolutely. For the alternating second series,

\frac{a _{n + 1}}{a _{n}} = 2 (\frac{n}{n + 1})^{5} ⟶ 2,

so it diverges. Alternating signs cannot rescue terms whose absolute values fail to tend to zero.

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