infinite continued fractions

Question 1

pk and qk relations:

Answer 1

pk/qk-p(k-1)/q(k-1)=(-1)^(k-1)/qkq(k-1) for all k>=1
pk/qk-p(k-2)/q(k-2)=(-1)^(k)xk/qkq(k-2) for all k>=2

Question 2

ck=pk/qk sequences:

Answer 2

c0<c2<…<c2k<… (increasing monotonically)
c1>c3>…>c(2k+1)>… (decreasing monotonically)
c0<c2<…<c(2k+1)<… (every odd element is greater than every even element)

Question 3

bounded monotone sequences converge:

Answer 3

let {cj} be a sequence that is monotonically increasing and bounded above by a real number, then it has a limit c that it’s converging to

Question 4

infinite simple continued fraction:

Answer 4

let {xk} be a sequence of integers with xk positive for all k>0 and pk,qk be the integers defined by the recurrence relations, the iscf corresponding to {xk} is [x0: x1,x2…]=lim(k->∞)[x0: x1,…,xk]=lim(k->∞)pk/qk. the nth convergent of [x0: x1,x2…] is the rational number pn/qn=[x0: x1,…,xn]

Question 5

|a-(pk/qk)|:

Answer 5

0<|a-(pk/qk)|<(1/qkq(k+1)), a=[x0:x1,…]

Question 6

if a real number has an iscf:

Answer 6

it’s irrational, this is an iff relation and the iscf is unique