Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 15 / b

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- Solution b

Solution

The generalized binomial series gives

(1 + x)^{1/3} = n = 0 \sum \infty u_{n} x^{n}, u_{0} = 1, u_{n} = \frac{( 1/3 ) ( 1/3 - 1 ) \dots ( 1/3 - n + 1 )}{n !} (n \geq 1) .

In particular,

(1 + x)^{1/3} = 1 + \frac{x}{3} - \frac{x ^{2}}{9} + \frac{5 x ^{3}}{81} - \frac{10 x ^{4}}{243} + \dots .

The recurrence

u_{n + 1} = u_{n} (1/3 - n) / (n + 1)

follows directly from the coefficient product. The power series has radius of convergence one and represents the real cube root for

∣ x ∣ < 1

.

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