72] LIMITS OF FUNCTIONS OF AN INTEGRAL VARIABLE 149 
12. If in addition the series ao + «i+«2 + ... is convergent, the series 
ao-\-air+a. 2 r i +... is convergent for 0<r^l, and its sum is not greater than 
the lesser of aQ- s r a i -\-a 2 +... and 1/(1 -r). 
13. The series 1 + T~2 + F2 _ 3 + "‘ 
is convergent. [For 1/(1.2...»)<l/2 ,l_1 .] 
14. The series 
1 _i ] i _—_4- i-i - 1 - u 
^1.2 1,2.3.4 ’ 1.2,31.2.3.4.5 
are convergent. 
15. The general harmonic series 
1 1 1 
a + a+ b + a+ 2b^~ 
where a and h are positive, diverges to + oo. 
[For u n = l/(a+nh) > l/{n (a + 6)}. Now compare with 1 -f (1/2) + (1/3) +....] 
16. Show that the series 
(«o - «i) + («i - «2) + («2 - «3) +.. • 
is convergent if and only if u n tends to a limit as n co . 
17. If «x + w 2 + «3 + ... is divergent, so is any series formed by grouping 
the terms in brackets in any way to form new single terms. 
18. Any series, formed by taking a selection of the terms of a convergent 
series of positive terms, is itself convergent. 
72. The representation of functions of a continuous 
real variable by means of limits. In the preceding sections 
we have frequently been concerned with limits such as 
lim <f> n {x), 
00 
and series such as 
tíi («) 4-u. 2 (cc)+ ... = lim {tii (x) + u 2 («)+... + u n (#)}, 
U-*’ OO 
in which the function of n whose limit we are seeking involves, 
besides n, another variable x. In such cases the limit is of course 
a function of x. Thus in § 69 we came across the function 
f{x) = lim n (y/x — 1): 
and the sum of the geometrical series 1 + x + x* + ... is a function 
of x, viz. the function which is equal to 1/(1— ¿e?) if — !<«<! and 
is undefined for all other values of x. 
Many of the apparently ‘ arbitrary ’ or ‘ unnatural ’ functions 
considered in Ch. II are capable of a simple representation of 
this kind, as will appear from the following examples.