If φ n s 1 s 2 s n then φ n φ n 1 s n and the

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[If φ ( n ) = s 1 + s 2 + · · · + s n then φ ( n ) - φ ( n - 1) = s n , and the theorem reduces to that proved in the last example.] 29. If s n = 1 2 { 1 - ( - 1) n } , so that s n is equal to 1 or 0 according as n is odd or even, then ( s 1 + s 2 + · · · + s n ) /n 1 2 as n → ∞ . [This example proves that the converse of Ex. 27 is not true: for s n oscillates as n → ∞ .] 30. If c n , s n denote the sums of the first n terms of the series 1 2 + cos θ + cos 2 θ + . . . , sin θ + sin 2 θ + . . . ,
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[IV : 88] LIMITS OF FUNCTIONS OF A 196 then lim( c 1 + c 2 + · · · + c n ) /n = 0 , lim( s 1 + s 2 + · · · + s n ) /n = 1 2 cot 1 2 θ.
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CHAPTER V LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS 89. Limits as x tends to . We shall now return to functions of a continuous real variable. We shall confine ourselves entirely to one-valued functions, * and we shall denote such a function by φ ( x ). We suppose x to assume successively all values corresponding to points on our fundamental straight line Λ, starting from some definite point on the line and progressing always to the right. In these circumstances we say that x tends to infinity , or to , and write x → ∞ . The only difference between the ‘tending of n to ’ discussed in the last chapter, and this ‘tending of x to ’, is that x assumes all values as it tends to , i.e. that the point P which corresponds to x coincides in turn with every point of Λ to the right of its initial position, whereas n tended to by a series of jumps. We can express this distinction by saying that x tends continuously to . As we explained at the beginning of the last chapter, there is a very close correspondence between functions of x and functions of n . Every function of n may be regarded as a selection from the values of a function of x . In the last chapter we discussed the peculiarities which may characterise the behaviour of a function φ ( n ) as n tends to . Now we are concerned with the same problem for a function φ ( x ); and the definitions and theorems to which we are led are practically repetitions of those of the last chapter. Thus corresponding to Def. 1 of § 58 we have: Definition 1. The function φ ( x ) is said to tend to the limit l as x tends to if, when any positive number , however small, is assigned, a number x 0 ( ) can be chosen such that, for all values of x equal to or greater than x 0 ( ) , φ ( x ) differs from l by less than , i.e. if | φ ( x ) - l | < when x = x 0 ( ) . * Thus x stands in this chapter for the one-valued function + x and not (as in § 26 ) for the two-valued function whose values are + x and - x . 197
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[V : 89] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 198 When this is the case we may write lim x →∞ φ ( x ) = l, or, when there is no risk of ambiguity, simply lim φ ( x ) = l , or φ ( x ) l . Similarly we have: Definition 2. The function φ ( x ) is said to tend to with x if, when any number Δ , however large, is assigned, we can choose a number x 0 (Δ) such that φ ( x ) > Δ when x = x 0 (Δ) .
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