LeGiang
05-12-2007, 07:12 AM
xin mời các cao thủ ra tay:
Let http://img359.images****.us/img359/5933/01up6.gif be an integer. Show that a real number x is rational if and only if the sequence http://img485.images****.us/img485/3466/01mq1.gif of digits of x in the expansion in basis b http://img412.images****.us/img412/7196/01rp6.gif
is ultimately periodic!Let http://img359.images****.us/img359/5933/01up6.gif be an integer. Let http://img469.images****.us/img469/7879/01ss2.gif be a bounded sequence of rational integers and http://img469.images****.us/img469/2613/01ks9.gif an increasing sequence of positive numbers. Assume there exists c>0 and http://img529.images****.us/img529/8151/01co0.gifsuch that, for allhttp://img465.images****.us/img465/213/01gi4.gif, http://img507.images****.us/img507/4563/01fj7.gif
a)deduce, for all http://img400.images****.us/img400/4476/01lt5.gif and http://img465.images****.us/img465/213/01gi4.gif, http://img209.images****.us/img209/6797/01jb3.gif
b)show that the number http://img520.images****.us/img520/6936/01rd7.gif is irrational if and only if the set http://img139.images****.us/img139/6198/01wo8.gif is infinite!
Let http://img359.images****.us/img359/5933/01up6.gif be an integer. Show that a real number x is rational if and only if the sequence http://img485.images****.us/img485/3466/01mq1.gif of digits of x in the expansion in basis b http://img412.images****.us/img412/7196/01rp6.gif
is ultimately periodic!Let http://img359.images****.us/img359/5933/01up6.gif be an integer. Let http://img469.images****.us/img469/7879/01ss2.gif be a bounded sequence of rational integers and http://img469.images****.us/img469/2613/01ks9.gif an increasing sequence of positive numbers. Assume there exists c>0 and http://img529.images****.us/img529/8151/01co0.gifsuch that, for allhttp://img465.images****.us/img465/213/01gi4.gif, http://img507.images****.us/img507/4563/01fj7.gif
a)deduce, for all http://img400.images****.us/img400/4476/01lt5.gif and http://img465.images****.us/img465/213/01gi4.gif, http://img209.images****.us/img209/6797/01jb3.gif
b)show that the number http://img520.images****.us/img520/6936/01rd7.gif is irrational if and only if the set http://img139.images****.us/img139/6198/01wo8.gif is infinite!