A Shrinking Family of Circles
Difficulty: Medium · Topics: Calculus, Series, Geometric Series
Kichrayib boruvchi aylanalar oilasi
For each natural number n, let C n be the circle with centre A n≤ft(dfrac14^ n-1,\ 0 ) that is tangent to the line 3x+4y=0, and let B n be the point where that line touches C n.
graph "x":[-0.95,2.25],"y":[-1.25,1.15],"xTicks":[],"yTicks":[],"label":"The circle Cₙ centred at Aₙ on the x-axis, tangent to the line 3x + 4y = 0 at Bₙ","items":["type":"line","from":[-0.75,0.5625],"to":[1.25,-0.9375],"ink":true,"type":"circle","at":[1,0],"r":0.6,"type":"line","from":[0,0],"to":[0.64,-0.48],"ink":true,"type":"point","at":[1,0],"label":"Aₙ","anchor":"ne","ink":true,"type":"point","at":[0.64,-0.48],"label":"Bₙ","anchor":"sw","ink":false,"type":"label","at":[1.62,0.72],"text":"Cₙ","type":"label","at":[1.55,-0.95],"text":"3x + 4y = 0"]
Writing L n=overlineOB n, what is the sum of the series Σ n=1^∞L n? (Here O is the origin.)
Har bir natural n son uchun C n — markazi A n≤ft(dfrac14^ n-1,\ 0 ) bo‘lgan va 3x+4y=0 to‘g‘ri chiziqqa urinadigan aylana, B n — shu to‘g‘ri chiziq C n ga urinadigan nuqta.
graph "x":[-0.95,2.25],"y":[-1.25,1.15],"xTicks":[],"yTicks":[],"label":"The circle Cₙ centred at Aₙ on the x-axis, tangent to the line 3x + 4y = 0 at Bₙ","items":["type":"line","from":[-0.75,0.5625],"to":[1.25,-0.9375],"ink":true,"type":"circle","at":[1,0],"r":0.6,"type":"line","from":[0,0],"to":[0.64,-0.48],"ink":true,"type":"point","at":[1,0],"label":"Aₙ","anchor":"ne","ink":true,"type":"point","at":[0.64,-0.48],"label":"Bₙ","anchor":"sw","ink":false,"type":"label","at":[1.62,0.72],"text":"Cₙ","type":"label","at":[1.55,-0.95],"text":"3x + 4y = 0"]
L n=overlineOB n deb belgilasak, Σ n=1^∞L n qatorning yig‘indisi nechaga teng? (Bu yerda O — koordinatalar boshi.)