HINTS FOR CWORK2: --------- Essay Revise the material on continued fractions, producing a one/two page summary (essential for revision anyway). What are the main theorems? Rank them in order of importance. Try to explain each theorem in words. Find a way of saying what is a continued fraction without a formula. You do not have to be too specific. Think in terms of a recursive construction, or of a sequence of rational fractions converging to a limit. Use the analogy with digits sequences. Describe some properties of SCFs, taking the above theorems as main source of material. For instance, explain how convergents behave, and why they are so good in approximating the limit. --------- Problem 1. (a) There is a sqrt(6) hidden everywhere. (b) Deal with the sqrt(10) and sqrt(79) terms separately. --------- Problem 3. Compute the floor of each expression. Then it is easy. --------- Problem 5. (a) Multiplication by 2 yields an undesired factor 2 at the numerator of the continued fraction. Transfer it at denominator. (b) Looking at case (a) with all "2"s replaced by "n"s should suggest the method as well as the answer. Clearing numerators will lead to a domino-type effect. --------- Problem 6. (b) Less than four decimal digits means less than 1000. You must use the notion of best approximant. --------- Problem 7. (b) This is the difference of two probabilities, as computed in part (a). (d) First find in some book the Taylor series of the natural logarithm. The answer will be the difference of two infinite sums as from part (c), which you must relate to such Taylor series.