12.1 Definitions and basic theorems 597we suppose that e = in/n, where in and n are integers, we can suppose that
n > 0 and put x = 1 in the formula
ex } {22
T-! :j!+to obtain
m=1+1+Zi 13i+ {'ni+(n+1)'+(n-}2)i+
and then multiply by ii! to obtain the formula
(^111)
M=N + n-I-1+(n+1)(n-h2)+(n+I)(n+2)(n+3)+
where M and N are integers. Thus the quantity Q defined by
_ (^111)
Q n+1 +(n+1)(n+2)+(n+1)(n-h2)(n+3)+
is the difference of two integers and must therefore be an integer. But
n-{-1< Q<n++(n+l)2+(n+l)3+
12
so Q cannot be an integer.
19 Give a reasonable definition setting forth conditions under which a given
series
p1+V2+ V3+ ...
of vectors in E3 is said to be convergent. Show that the series will be convergent
if the series
Iv,I + IV21 + Iv31 +
is convergent.
20 This problem involves rearrangements of series of nonnegative terms.
Let uk? 0 for each k and let
S = nl+ U2 + U3 + U4 + ...-
Let m1, m2i m3,. be a sequence of positive integers, not necessarily in their
natural order, in which each positive integer appears exactly once. Prove that
.r = 'U-1 + Ums + Ums + Um4 +
Hint: Let
to = U.1+Urn,+ +Ur,,.
Show that t 5 r and that if e > 0, then t > s - e whenever n is sufficiently
great.
21 This problem has a preamble. To pour acid upon the idea that each
rearrangement of the series
(1) 1-+g-a+a-8+T-a+.