Pk = Pk-i&k + Pk-2, qk = qk-iak + qk-2 with fc = 2,3,...,n,
and define
^0 = 00 , i?fc = « o + i h i h ... + i , k = l,2,...,n .
(jRfc is called the kth convergent to ao + T^ + T^ + --- + -n~J-
Show that
Rk = for k 0, l,...,n.
1.1.19. Show that if pk.qk are defined as in the foregoing problem
and the ao,oi,...,a
n
are integers, then
Pk-iQk ~ qk-iPk =
(~l)fc
for k = 1,2,..., n.
Use this equality to conclude that p/- and qk are co-prime.
1.1.20. For an irrational number x we define a sequence {xn} by
1 1 1
# i =
r^5
x
2
x - [ x ] ' x i - [ x i ] ' " * ' n xn-i - [ x
n
_ i ] ' " "
Moreover, we put ao = [x], an = [xn], n = 1,2,..., and
Rn = a0 +
T-
1
+ -r-1 + ... + r 1 -.
|ai |a2 |an
Show that the difference between the number x and its nth convergent
is given by
x
- R = v ;
where pn,7n are defined in 1.1.18. Conclude that x is between its
two consecutive convergents.
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