Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 19 / c

Table of contents
- Solution c
- i c
  - Solution i
- ii c
  - Solution ii

Solution

For a series with eventually nonzero terms

a_{n}

, suppose

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

exists. The ratio test says that

r < 1

implies absolute convergence, while

r > 1

implies divergence because the terms fail to tend to zero. At

r = 1

the test is inconclusive.

i

Solution

Here

∣ a_{n + 1} / a_{n} ∣ = n / (n + 1) \to 1

. Therefore the ratio test is inconclusive. The series does in fact converge by the alternating series test, and fails to converge absolutely because the harmonic series diverges; these facts use other tests.

ii

Solution

For the positive summand

a_{n} = (n!)^{3} e^{3 n} / (3 n)!

, the ratio test gives

\frac{a _{n + 1}}{a _{n}} = \frac{e ^{3} ( n + 1 ) ^{3}}{( 3 n + 1 ) ( 3 n + 2 ) ( 3 n + 3 )} ⟶ \frac{e ^{3}}{27} < 1.

Thus the series converges absolutely.

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