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.