3) Evaluation of Infinite Simple Continued Fractions Flashcards
What is an infinite simple continued fraction
What are the key identities involving the convergents pk/qk of a continued fraction
What does the Bounded Monotone Sequence Theorem state
Let (ck)k≥0 be a sequence of real numbers that is monotonically increasing and bounded above by a real number M. Then the sequence (ck)k≥0 has a limit c, with c ≤ M
What does the theorem about the convergence of continued fraction convergents state
Let {xk}k≥0 be an infinite sequence of integers, with xk positive for all k > 0 and let pk, qk be the integers defined by the recurrence relations (2.2). Then the sequence of rationals ck =pk/qconverges to a real number
How is the value of an infinite simple continued fraction defined, and what is a convergent
What is the error bound for the kth convergent of an infinite simple continued fraction
How can the reciprocal of an infinite simple continued fraction be expressed as another continued fraction
How can an infinite simple continued fraction be split at an index n
What is a periodic simple continued fraction
What is a quadratic irrational
A quadratic irrational is an irrational real number that is a solution of a quadratic polynomial with rational coefficients
How can a quadratic irrational be characterized
Let α ∈ R. Then α is a quadratic irrational if and only if there exist r, s ∈ Q with s ̸= 0 and a positive integer d that is not a perfect square such that α = r + s√d
What is a purely periodic continued fraction
What is the conjugate of a quadratic irrational
Let α = r + s√d be a quadratic irrational. We define the conjugate of α to be the quadratic irrational α∗ = r − s√d
What is a reduced quadratic irrational
A quadratic irrational α is reduced if α > 1 and −1 < α∗ < 0.
What is the continued fraction representation of d for a non-square integer d
How do continued fraction properties correspond to different types of numbers
