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# 5 the series 1 z 1 z z 1 z 2 converges to the sum 1 1

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5. The series 1 + z 1 + z + z 1 + z 2 + . . . converges to the sum 1 1 - z 1 + z = 1+ z if | z/ (1+ z ) | < 1. Show that this condition is equivalent to the condition that z has a real part greater than - 1 2 .

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[IV : 88] LIMITS OF FUNCTIONS OF A 190 MISCELLANEOUS EXAMPLES ON CHAPTER IV. 1. The function φ ( n ) takes the values 1, 0, 0, 0, 1, 0, 0, 0, 1, . . . when n = 0, 1, 2, . . . . Express φ ( n ) in terms of n by a formula which does not involve trigonometrical functions. [ φ ( n ) = 1 4 { 1 + ( - 1) n + i n + ( - i ) n } .] 2. If φ ( n ) steadily increases, and ψ ( n ) steadily decreases, as n tends to , and if ψ ( n ) > φ ( n ) for all values of n , then both φ ( n ) and ψ ( n ) tend to limits, and lim φ ( n ) 5 lim ψ ( n ). [This is an immediate corollary from § 69 .] 3. Prove that, if φ ( n ) = 1 + 1 n n , ψ ( n ) = 1 - 1 n - n , then φ ( n + 1) > φ ( n ) and ψ ( n + 1) < ψ ( n ). [The first result has already been proved in § 73 .] 4. Prove also that ψ ( n ) > φ ( n ) for all values of n : and deduce (by means of the preceding examples) that both φ ( n ) and ψ ( n ) tend to limits as n tends to . * 5. The arithmetic mean of the products of all distinct pairs of positive integers whose sum is n is denoted by S n . Show that lim( S n /n 2 ) = 1 / 6. ( Math. Trip. 1903.) 6. Prove that if x 1 = 1 2 { x + ( A/x ) } , x 2 = 1 2 { x 1 + ( A/x 1 ) } , and so on, x and A being positive, then lim x n = A . [Prove first that x n - A x n + A = x - A x + A 2 n .] 7. If φ ( n ) is a positive integer for all values of n , and tends to with n , then x φ ( n ) tends to 0 if 0 < x < 1 and to + if x > 1. Discuss the behaviour of x φ ( n ) , as n → ∞ , for other values of x . 8. If a n increases or decreases steadily as n increases, then the same is true of ( a 1 + a 2 + · · · + a n ) /n . 9. If x n +1 = k + x n , and k and x 1 are positive, then the sequence x 1 , x 2 , x 3 , . . . is an increasing or decreasing sequence according as x 1 is less than or * A proof that lim { ψ ( n ) - φ ( n ) } = 0, and that therefore each function tends to the limit e , will be found in Chrystal’s Algebra , vol. ii, p. 78. We shall however prove this in Ch. IX by a different method. Exs. 8–12 are taken from Bromwich’s Infinite Series .
[IV : 88] POSITIVE INTEGRAL VARIABLE 191 greater than α , the positive root of the equation x 2 = x + k ; and in either case x n α as n → ∞ . 10. If x n +1 = k/ (1 + x n ), and k and x 1 are positive, then the sequences x 1 , x 3 , x 5 , . . . and x 2 , x 4 , x 6 , . . . are one an increasing and the other a decreas- ing sequence, and each sequence tends to the limit α , the positive root of the equation x 2 + x = k . 11. The function f ( x ) is increasing and continuous (see Ch. V ) for all values of x , and a sequence x 1 , x 2 , x 3 , . . . is defined by the equation x n +1 = f ( x n ). Discuss on general graphical grounds the question as to whether x n tends to a root of the equation x = f ( x ). Consider in particular the case in which this equation has only one root, distinguishing the cases in which the curve y = f ( x ) crosses the line y = x from above to below and from below to above.

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