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❆♣♣✳ ❆✿ ❙❡q✉❡♥❝❡s ❛♥❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥ ✶ , ✷ ❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r② ✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s ✷ ❆✉❣ ✷✶✱ ✷✵✶✻

❙❡q✉❡♥❝❡s ❙❡q✉❡♥❝❡s ✐s ❛ ❝❡♥tr❛❧ t♦♣✐❝ ✐♥ ♠❛t❤❡♠❛t✐❝s✿ x ✵ , x ✶ , x ✷ , . . . , x n , . . . , ❊①❛♠♣❧❡✿ ❛❧❧ ♦❞❞ ♥✉♠❜❡rs ✶ , ✸ , ✺ , ✼ , . . . , ✷ n + ✶ , . . . ❋♦r t❤✐s s❡q✉❡♥❝❡ ✇❡ ❤❛✈❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ n ✲t❤ t❡r♠✿ x n = ✷ n + ✶ ❛♥❞ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ s❡q✉❡♥❝❡ ♠♦r❡ ❝♦♠♣❛❝t❧② ❛s ( x n ) ∞ n = ✵ , x n = ✷ n + ✶

❖t❤❡r ❡①❛♠♣❧❡s ♦❢ s❡q✉❡♥❝❡s n = ✵ , x n = n ✷ ✶ , ✹ , ✾ , ✶✻ , ✷✺ , . . . ( x n ) ∞ ✶ , ✶ ✷ , ✶ ✸ , ✶ ✶ ( x n ) ∞ ✹ , . . . n = ✵ , x n = n + ✶ ( x n ) ∞ ✶ , ✶ , ✷ , ✻ , ✷✹ , . . . n = ✵ , x n = n ! n x j ✶ , ✶ + x , ✶ + x + ✶ ✷ x ✷ , ✶ + x + ✶ ✷ x ✷ + ✶ � ✻ x ✸ , . . . ( x n ) ∞ n = ✵ , x n = j ! j = ✵

❋✐♥✐t❡ ❛♥❞ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s ■♥✜♥✐t❡ s❡q✉❡♥❝❡s ❤❛✈❡ ❛♥ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ t❡r♠s ✭ n → ∞ ✮ ■♥ ♠❛t❤❡♠❛t✐❝s✱ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ■♥ r❡❛❧✲❧✐❢❡ ❛♣♣❧✐❝❛t✐♦♥s✱ s❡q✉❡♥❝❡s ❛r❡ ✉s✉❛❧❧② ✜♥✐t❡✿ ( x n ) N n = ✵ ❊①❛♠♣❧❡✿ ♥✉♠❜❡r ♦❢ ❛♣♣r♦✈❡❞ ❡①❡r❝✐s❡s ❡✈❡r② ✇❡❡❦ ✐♥ ■◆❋✶✶✵✵ x ✵ , x ✶ , x ✷ , . . . , x ✶✺ ❊①❛♠♣❧❡✿ t❤❡ ❛♥♥✉❛❧ ✈❛❧✉❡ ♦❢ ❛ ❧♦❛♥ x ✵ , x ✶ , . . . , x ✷✵

❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❋♦r s❡q✉❡♥❝❡s ♦❝❝✉r✐♥❣ ✐♥ ♠♦❞❡❧✐♥❣ ♦❢ r❡❛❧✲✇♦r❧❞ ♣❤❡♥♦♠❡♥❛✱ t❤❡r❡ ✐s s❡❧❞♦♠ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ n ✲t❤ t❡r♠ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦❢t❡♥ s❡t ✉♣ ♦♥❡ ♦r ♠♦r❡ ❡q✉❛t✐♦♥s ❣♦✈❡r♥✐♥❣ t❤❡ s❡q✉❡♥❝❡ ❙✉❝❤ ❡q✉❛t✐♦♥s ❛r❡ ❝❛❧❧❡❞ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❲✐t❤ ❛ ❝♦♠♣✉t❡r ✐t ✐s t❤❡♥ ✈❡r② ❡❛s② t♦ ❣❡♥❡r❛t❡ t❤❡ s❡q✉❡♥❝❡ ❜② s♦❧✈✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❤❛✈❡ ❧♦ts ♦❢ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❛r❡ ✈❡r② ❡❛s② t♦ s♦❧✈❡ ♦♥ ❛ ❝♦♠♣✉t❡r✱ ❜✉t ♦❢t❡♥ ❝♦♠♣❧✐❝❛t❡❞ ♦r ✐♠♣♦ss✐❜❧❡ t♦ s♦❧✈❡ ❢♦r x n ✭❛s ❛ ❢♦r♠✉❧❛✮ ❜② ♣❡♥ ❛♥❞ ♣❛♣❡r✦ ❚❤❡ ♣r♦❣r❛♠s r❡q✉✐r❡ ♦♥❧② ❧♦♦♣s ❛♥❞ ❛rr❛②s

▼♦❞❡❧✐♥❣ ✐♥t❡r❡st r❛t❡s Pr♦❜❧❡♠✿ P✉t x ✵ ♠♦♥❡② ✐♥ ❛ ❜❛♥❦ ❛t ②❡❛r ✵✳ ❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ❛❢t❡r N ②❡❛rs ✐❢ t❤❡ ✐♥t❡r❡st r❛t❡ ✐s p ♣❡r❝❡♥t ♣❡r ②❡❛r❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♥❢♦r♠❛t✐♦♥ r❡❧❛t❡s t❤❡ ✈❛❧✉❡ ❛t ②❡❛r n ✱ x n ✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ x n − ✶ ✿ p x n = x n − ✶ + ✶✵✵ x n − ✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r x n ❄ ❙t❛rt ✇✐t❤ x ✵ ✱ ❝♦♠♣✉t❡ x ✶ ✱ x ✷ ✱ ✳✳✳

▼♦❞❡❧✐♥❣ ✐♥t❡r❡st r❛t❡s Pr♦❜❧❡♠✿ P✉t x ✵ ♠♦♥❡② ✐♥ ❛ ❜❛♥❦ ❛t ②❡❛r ✵✳ ❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ❛❢t❡r N ②❡❛rs ✐❢ t❤❡ ✐♥t❡r❡st r❛t❡ ✐s p ♣❡r❝❡♥t ♣❡r ②❡❛r❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♥❢♦r♠❛t✐♦♥ r❡❧❛t❡s t❤❡ ✈❛❧✉❡ ❛t ②❡❛r n ✱ x n ✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ x n − ✶ ✿ p x n = x n − ✶ + ✶✵✵ x n − ✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r x n ❄ ❙t❛rt ✇✐t❤ x ✵ ✱ ❝♦♠♣✉t❡ x ✶ ✱ x ✷ ✱ ✳✳✳

▼♦❞❡❧✐♥❣ ✐♥t❡r❡st r❛t❡s Pr♦❜❧❡♠✿ P✉t x ✵ ♠♦♥❡② ✐♥ ❛ ❜❛♥❦ ❛t ②❡❛r ✵✳ ❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ❛❢t❡r N ②❡❛rs ✐❢ t❤❡ ✐♥t❡r❡st r❛t❡ ✐s p ♣❡r❝❡♥t ♣❡r ②❡❛r❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♥❢♦r♠❛t✐♦♥ r❡❧❛t❡s t❤❡ ✈❛❧✉❡ ❛t ②❡❛r n ✱ x n ✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ x n − ✶ ✿ p x n = x n − ✶ + ✶✵✵ x n − ✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r x n ❄ ❙t❛rt ✇✐t❤ x ✵ ✱ ❝♦♠♣✉t❡ x ✶ ✱ x ✷ ✱ ✳✳✳

❙✐♠✉❧❛t✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❢♦r ✐♥t❡r❡st r❛t❡s ❲❤❛t ❞♦❡s ✐t ♠❡❛♥ t♦ s✐♠✉❧❛t❡❄ ❙♦❧✈❡ ♠❛t❤ ❡q✉❛t✐♦♥s ❜② r❡♣❡❛t✐♥❣ ❛ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ ✭r❡❧❛t✐♦♥✮ ♠❛♥② t✐♠❡s ✭❜♦r✐♥❣✱ ❜✉t ✇❡❧❧ s✉✐t❡❞ ❢♦r ❛ ❝♦♠♣✉t❡r✦✮ Pr♦❣r❛♠ ❢♦r x n = x n − ✶ + ( p / ✶✵✵ ) x n − ✶ ✿ ❢r♦♠ s❝✐t♦♦❧s✳st❞ ✐♠♣♦rt ✯ ①✵ ❂ ✶✵✵ ★ ✐♥✐t✐❛❧ ❛♠♦✉♥t ♣ ❂ ✺ ★ ✐♥t❡r❡st r❛t❡ ◆ ❂ ✹ ★ ♥✉♠❜❡r ♦❢ ②❡❛rs ✐♥❞❡①❴s❡t ❂ r❛♥❣❡✭◆✰✶✮ ① ❂ ③❡r♦s✭❧❡♥✭✐♥❞❡①❴s❡t✮✮ ★ ❙♦❧✉t✐♦♥✿ ①❬✵❪ ❂ ①✵ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ✭♣✴✶✵✵✳✵✮✯①❬♥✲✶❪ ♣r✐♥t ① ♣❧♦t✭✐♥❞❡①❴s❡t✱ ①✱ ✬r♦✬✱ ①❧❛❜❡❧❂✬②❡❛rs✬✱ ②❧❛❜❡❧❂✬❛♠♦✉♥t✬✮

❲❡ ❞♦ ♥♦t ♥❡❡❞ t♦ st♦r❡ t❤❡ ❡♥t✐r❡ s❡q✉❡♥❝❡✱ ❜✉t ✐t ✐s ❝♦♥✈❡♥✐❡♥t ❢♦r ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ ❧❛t❡r ♣❧♦tt✐♥❣ Pr❡✈✐♦✉s ♣r♦❣r❛♠ st♦r❡s ❛❧❧ t❤❡ x n ✈❛❧✉❡s ✐♥ ❛ ◆✉♠P② ❛rr❛② ❚♦ ❝♦♠♣✉t❡ x n ✱ ✇❡ ♦♥❧② ♥❡❡❞ ♦♥❡ ♣r❡✈✐♦✉s ✈❛❧✉❡✱ x n − ✶ ❚❤✉s✱ ✇❡ ❝♦✉❧❞ ♦♥❧② st♦r❡ t❤❡ t✇♦ ❧❛st ✈❛❧✉❡s ✐♥ ♠❡♠♦r②✿ ①❴♦❧❞ ❂ ①✵ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❴♥❡✇ ❂ ①❴♦❧❞ ✰ ✭♣✴✶✵✵✳✮✯①❴♦❧❞ ①❴♦❧❞ ❂ ①❴♥❡✇ ★ ①❴♥❡✇ ❜❡❝♦♠❡s ①❴♦❧❞ ❛t ♥❡①t st❡♣ ❍♦✇❡✈❡r✱ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ❛♥ ❛rr❛② ①❬♥❪ ✐s s✐♠♣❧❡r✱ s❛❢❡r✱ ❛♥❞ ❡♥❛❜❧❡s ♣❧♦tt✐♥❣ t❤❡ s❡q✉❡♥❝❡✱ s♦ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ ❛rr❛②s ✐♥ t❤❡ ❡①❛♠♣❧❡s

❉❛✐❧② ✐♥t❡r❡st r❛t❡ ❆ ♠♦r❡ r❡❧❡✈❛♥t ♠♦❞❡❧ ✐s t♦ ❛❞❞ t❤❡ ✐♥t❡r❡st ❡✈❡r② ❞❛② ❚❤❡ ✐♥t❡r❡st r❛t❡ ♣❡r ❞❛② ✐s r = p / D ✐❢ p ✐s t❤❡ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡ ❛♥❞ D ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ✐♥ ❛ ②❡❛r ❆ ❝♦♠♠♦♥ ♠♦❞❡❧ ✐♥ ❜✉s✐♥❡ss ❛♣♣❧✐❡s D = ✸✻✵✱ ❜✉t n ❝♦✉♥ts ❡①❛❝t ✭❛❧❧✮ ❞❛②s ❏✉st ❛ ♠✐♥♦r ❝❤❛♥❣❡ ✐♥ t❤❡ ♠♦❞❡❧✿ r x n = x n − ✶ + ✶✵✵ x n − ✶ ❍♦✇ ❝❛♥ ✇❡ ✜♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❜❡t✇❡❡♥ t✇♦ ❞❛t❡s❄ ❃❃❃ ✐♠♣♦rt ❞❛t❡t✐♠❡ ❃❃❃ ❞❛t❡✶ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✵✼✱ ✽✱ ✸✮ ★ ❆✉❣ ✸✱ ✷✵✵✼ ❃❃❃ ❞❛t❡✷ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✵✽✱ ✽✱ ✹✮ ★ ❆✉❣ ✹✱ ✷✵✵✽ ❃❃❃ ❞✐❢❢ ❂ ❞❛t❡✷ ✲ ❞❛t❡✶ ❃❃❃ ♣r✐♥t ❞✐❢❢✳❞❛②s ✸✻✼