Reduction formulae of the type are iteration formulae – and, like other iteration formulae, when they are used, attention must be paid to their numerical properties. This is illustrated by considering the integral

Evaluate I₀ on your calculator and use the reduction formula to calculate In, n = 1, 2, . . . . , 10. Since

Compare this with your results, and explain the discrepancy. Since the iteration diverges when used for n increasing (that is, on setting n = 1, 2, 3, . . . . in turn),  it will converge when used for n decreasing (say n = 50, 49, 48, . . . ). Since 0 19 10. Continue the iteration backwards to find, eventually, I₀.

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"I ] = Prove that x"edx I = 1nl-i (n > 0) 0 with I, 1 e . -