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2 θπ sin nθπ sin 1 2 θπ must be numerically

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2 θπ - sin nθπ sin 1 2 θπ must be numerically less than / | sin 1 2 θπ | . Similarly cos( n - 1 2 ) θπ = cos nθπ cos 1 2 θπ + sin nθπ sin 1 2 θπ must be numerically less than / | sin 1 2 θπ | ; and so each of cos nθπ cos 1 2 θπ , sin nθπ sin 1 2 θπ must be numerically less than / | sin 1 2 θπ | . That is to say, cos nθπ cos 1 2 θπ is very small if n is large, and this can only be the case if cos nθπ is very small. Similarly sin nθπ must be very small, so that l must be zero. But it is impossible that cos nθπ and sin nθπ can both be very small, as the sum of their squares is unity. Thus the hypothesis that sin nθπ tends to a limit l is impossible, and therefore sin nθπ oscillates as n tends to . The reader should consider with particular care the argument ‘cos nθπ cos 1 2 θπ is very small, and this can only be the case if cos nθπ is very small’. Why, he may ask, should it not be the other factor cos 1 2 θπ which is ‘very small’? The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘ φ ( n ) is very small’ for large values of n , we mean that we can choose n 0 so that φ ( n ) is numerically smaller than any assigned number, if n = n 0 . Such an assertion is palpably absurd when made of a fixed number such as cos 1 2 θπ , which is not zero. Prove similarly that cos nθπ oscillates finitely, unless θ is an even integer. 8. sin nθπ + (1 /n ), sin nθπ + 1, sin nθπ + n , ( - 1) n sin nθπ . 9. a cos nθπ + b sin nθπ , sin 2 nθπ , a cos 2 nθπ + b sin 2 nθπ . 10. a + bn + ( - 1) n ( c + dn ) + e cos nθπ + f sin nθπ . 11. n sin nθπ . If θ is integral, then φ ( n ) = 0, φ ( n ) 0. If θ is rational but not integral, or irrational, then φ ( n ) oscillates infinitely. 12. n ( a cos 2 nθπ + b sin 2 nθπ ). In this case φ ( n ) tends to + if a and b are both positive, but to -∞ if both are negative. Consider the special cases in which a = 0, b > 0, or a > 0, b = 0, or a = 0, b = 0. If a and b have opposite signs φ ( n ) generally oscillates infinitely. Consider any exceptional cases. 13. sin( n 2 θπ ). If θ is integral, then φ ( n ) 0. Otherwise φ ( n ) oscillates finitely, as may be shown by arguments similar to though more complex than those used in Exs. xxiii . 9 and xxiv . 7. * * See Bromwich’s Infinite Series , p. 485.

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[IV : 62] LIMITS OF FUNCTIONS OF A 148 14. sin( n ! θπ ). If θ has a rational value p/q , then n ! θ is certainly integral for all values of n greater than or equal to q . Hence φ ( n ) 0. The case in which θ is irrational cannot be dealt with without the aid of considerations of a much more difficult character. 15. cos( n ! θπ ), a cos 2 ( n ! θπ ) + b sin 2 ( n ! θπ ), where θ is rational. 16. an - [ bn ], ( - 1) n ( an - [ bn ]). 17. [ n ], ( - 1) n [ n ], n - [ n ]. 18. The smallest prime factor of n . When n is a prime, φ ( n ) = n . When n is even, φ ( n ) = 2. Thus φ ( n ) oscillates infinitely.
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