qs r - - PowerPoint PPT Presentation

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

n=✵,

xn = ✷n + ✶

❖t❤❡r ❡①❛♠♣❧❡s ♦❢ s❡q✉❡♥❝❡s

✶, ✹, ✾, ✶✻, ✷✺, . . . (xn)∞

n=✵, xn = n✷

✶, ✶ ✷, ✶ ✸, ✶ ✹, . . . (xn)∞

n=✵, xn =

✶ n + ✶ ✶, ✶, ✷, ✻, ✷✹, . . . (xn)∞

n=✵, xn = n!

✶, ✶+x, ✶+x + ✶ ✷x✷, ✶+x + ✶ ✷x✷+ ✶ ✻x✸, . . . (xn)∞

n=✵, xn = n

• j=✵

xj 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❡✿ (xn)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 xn ✭❛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✱ xn✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ xn−✶✿ xn = xn−✶ + p ✶✵✵xn−✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r xn❄ ❙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✱ xn✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ xn−✶✿ xn = xn−✶ + p ✶✵✵xn−✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r xn❄ ❙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✱ xn✱ t♦ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✱ xn−✶✿ xn = xn−✶ + p ✶✵✵xn−✶ ❍♦✇ t♦ s♦❧✈❡ ❢♦r xn❄ ❙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 xn = xn−✶ + (p/✶✵✵)xn−✶✿

❢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❤❡ xn ✈❛❧✉❡s ✐♥ ❛ ◆✉♠P② ❛rr❛② ❚♦ ❝♦♠♣✉t❡ xn✱ ✇❡ ♦♥❧② ♥❡❡❞ ♦♥❡ ♣r❡✈✐♦✉s ✈❛❧✉❡✱ xn−✶ ❚❤✉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❤❡ ♠♦❞❡❧✿ xn = xn−✶ + r ✶✵✵xn−✶ ❍♦✇ ❝❛♥ ✇❡ ✜♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ❜❡t✇❡❡♥ t✇♦ ❞❛t❡s❄

❃❃❃ ✐♠♣♦rt ❞❛t❡t✐♠❡ ❃❃❃ ❞❛t❡✶ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✵✼✱ ✽✱ ✸✮ ★ ❆✉❣ ✸✱ ✷✵✵✼ ❃❃❃ ❞❛t❡✷ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✵✽✱ ✽✱ ✹✮ ★ ❆✉❣ ✹✱ ✷✵✵✽ ❃❃❃ ❞✐❢❢ ❂ ❞❛t❡✷ ✲ ❞❛t❡✶ ❃❃❃ ♣r✐♥t ❞✐❢❢✳❞❛②s ✸✻✼

Pr♦❣r❛♠ ❢♦r ❞❛✐❧② ✐♥t❡r❡st r❛t❡

❢r♦♠ s❝✐t♦♦❧s✳st❞ ✐♠♣♦rt ✯ ①✵ ❂ ✶✵✵ ★ ✐♥✐t✐❛❧ ❛♠♦✉♥t ♣ ❂ ✺ ★ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡ r ❂ ♣✴✸✻✵✳✵ ★ ❞❛✐❧② ✐♥t❡r❡st r❛t❡ ✐♠♣♦rt ❞❛t❡t✐♠❡ ❞❛t❡✶ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✵✼✱ ✽✱ ✸✮ ❞❛t❡✷ ❂ ❞❛t❡t✐♠❡✳❞❛t❡✭✷✵✶✶✱ ✽✱ ✸✮ ❞✐❢❢ ❂ ❞❛t❡✷ ✲ ❞❛t❡✶ ◆ ❂ ❞✐❢❢✳❞❛②s ✐♥❞❡①❴s❡t ❂ r❛♥❣❡✭◆✰✶✮ ① ❂ ③❡r♦s✭❧❡♥✭✐♥❞❡①❴s❡t✮✮ ★ ❙♦❧✉t✐♦♥✿ ①❬✵❪ ❂ ①✵ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ✭r✴✶✵✵✳✵✮✯①❬♥✲✶❪ ♣r✐♥t ① ♣❧♦t✭✐♥❞❡①❴s❡t✱ ①✱ ✬r♦✬✱ ①❧❛❜❡❧❂✬❞❛②s✬✱ ②❧❛❜❡❧❂✬❛♠♦✉♥t✬✮

❇✉t t❤❡ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡ ♠❛② ❝❤❛♥❣❡ q✉✐t❡ ♦❢t❡♥✳✳✳

❱❛r②✐♥❣ p ♠❡❛♥s pn✿ ❈♦✉❧❞ ♥♦t ❜❡ ❤❛♥❞❧❡❞ ✐♥ s❝❤♦♦❧ ✭❝❛♥♥♦t ❛♣♣❧② xn = x✵(✶ +

p ✶✵✵)n✮

❆ ✈❛r②✐♥❣ p ❝❛✉s❡s ♥♦ ♣r♦❜❧❡♠s ✐♥ t❤❡ ♣r♦❣r❛♠✿ ❥✉st ✜❧❧ ❛♥ ❛rr❛② ♣ ✇✐t❤ ❝♦rr❡❝t ✐♥t❡r❡st r❛t❡ ❢♦r ❞❛② ♥ ▼♦❞✐✜❡❞ ♣r♦❣r❛♠✿

♣ ❂ ③❡r♦s✭❧❡♥✭✐♥❞❡①❴s❡t✮✮ ★ ❢✐❧❧ ♣❬♥❪ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t ✭♠✐❣❤t ❜❡ ♥♦♥✲tr✐✈✐❛❧✳✳✳✮ r ❂ ♣✴✸✻✵✳✵ ★ ❞❛✐❧② ✐♥t❡r❡st r❛t❡ ① ❂ ③❡r♦s✭❧❡♥✭✐♥❞❡①❴s❡t✮✮ ①❬✵❪ ❂ ①✵ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ✭r❬♥✲✶❪✴✶✵✵✳✵✮✯①❬♥✲✶❪

P❛②❜❛❝❦ ♦❢ ❛ ❧♦❛♥

❆ ❧♦❛♥ L ✐s ♣❛✐❞ ❜❛❝❦ ✇✐t❤ ❛ ✜①❡❞ ❛♠♦✉♥t L/N ❡✈❡r② ♠♦♥t❤ ♦✈❡r N ♠♦♥t❤s ✰ t❤❡ ✐♥t❡r❡st r❛t❡ ♦❢ t❤❡ ❧♦❛♥ p✿ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡✱ p/✶✷ : ♠♦♥t❤❧② r❛t❡ ▲❡t xn ❜❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❧♦❛♥ ❛t t❤❡ ❡♥❞ ♦❢ ♠♦♥t❤ n ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥ ❢r♦♠ ♦♥❡ ♠♦♥t❤ t♦ t❤❡ t❡①t✿ xn = xn−✶ + p ✶✷ · ✶✵✵xn−✶ − ( p ✶✷ · ✶✵✵xn−✶ + L N ) ✇❤✐❝❤ s✐♠♣❧✐✜❡s t♦ xn = xn−✶ − L N ✭L/N ♠❛❦❡s t❤❡ ❡q✉❛t✐♦♥ ♥♦♥❤♦♠♦❣❡♥❡♦✉s✮

❍♦✇ t♦ ♠❛❦❡ ❛ ❧✐✈✐♥❣ ❢r♦♠ ❛ ❢♦rt✉♥❡ ✇✐t❤ ❝♦♥st❛♥t ❝♦♥s✉♠♣t✐♦♥

❲❡ ❤❛✈❡ ❛ ❢♦rt✉♥❡ F ✐♥✈❡st❡❞ ✇✐t❤ ❛♥ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡ ♦❢ p ♣❡r❝❡♥t ❊✈❡r② ②❡❛r ✇❡ ♣❧❛♥ t♦ ❝♦♥s✉♠❡ ❛♥ ❛♠♦✉♥t cn ✭n ❝♦✉♥ts ②❡❛rs✮ ▲❡t xn ❜❡ ♦✉r ❢♦rt✉♥❡ ❛t ②❡❛r n ❆ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥ ❢r♦♠ ♦♥❡ ②❡❛r t♦ t❤❡ ♦t❤❡r ✐s xn = xn−✶ + p ✶✵✵xn−✶ − cn ❙✐♠♣❧❡st ♣♦ss✐❜✐❧✐t②✿ ❦❡❡♣ cn ❝♦♥st❛♥t✱ ❜✉t ✐♥✢❛t✐♦♥ ❞❡♠❛♥❞s cn t♦ ✐♥❝r❡❛s❡✳✳✳

❍♦✇ t♦ ♠❛❦❡ ❛ ❧✐✈✐♥❣ ❢r♦♠ ❛ ❢♦rt✉♥❡ ✇✐t❤ ✐♥✢❛t✐♦♥✲❛❞❥✉st❡❞ ❝♦♥s✉♠♣t✐♦♥

❆ss✉♠❡ I ♣❡r❝❡♥t ✐♥✢❛t✐♦♥ ♣❡r ②❡❛r ❙t❛rt ✇✐t❤ c✵ ❛s q ♣❡r❝❡♥t ♦❢ t❤❡ ✐♥t❡r❡st t❤❡ ✜rst ②❡❛r cn t❤❡♥ ❞❡✈❡❧♦♣s ❛s ♠♦♥❡② ✇✐t❤ ✐♥t❡r❡st r❛t❡ I xn ❞❡✈❡❧♦♣s ✇✐t❤ r❛t❡ p ❜✉t ✇✐t❤ ❛ ❧♦ss cn ❡✈❡r② ②❡❛r✿ xn = xn−✶ + p ✶✵✵xn−✶ − cn−✶, x✵ = F, c✵ = pq ✶✵✹ F cn = cn−✶ + I ✶✵✵cn−✶ ❚❤✐s ✐s ❛ ❝♦✉♣❧❡❞ s②st❡♠ ♦❢ t✇♦ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ❜✉t t❤❡ ♣r♦❣r❛♠♠✐♥❣ ✐s st✐❧❧ s✐♠♣❧❡✿ ✇❡ ✉♣❞❛t❡ t✇♦ ❛rr❛②s✱ ✜rst ①❬♥❪✱ t❤❡♥ ❝❬♥❪✱ ✐♥s✐❞❡ t❤❡ ❧♦♦♣ ✭❣♦♦❞ ❡①❡r❝✐s❡✦✮

❚❤❡ ♠❛t❤❡♠❛t✐❝s ♦❢ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs

◆♦ ♣r♦❣r❛♠♠✐♥❣ ♦r ♠❛t❤ ❝♦✉rs❡ ✐s ❝♦♠♣❧❡t❡ ✇✐t❤♦✉t ❛♥ ❡①❛♠♣❧❡ ♦♥ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs✿ xn = xn−✶ + xn−✷, x✵ = ✶, x✶ = ✶ ▼❛t❤❡♠❛t✐❝❛❧ ❝❧❛ss✐✜❝❛t✐♦♥ ❚❤✐s ✐s ❛ ❤♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r ✭s❡❝♦♥❞ ♦r❞❡r ♠❡❛♥s t❤r❡❡ ❧❡✈❡❧s✿ n✱ n − ✶✱ n − ✷✮✳ ❚❤✐s ❝❧❛ss✐✜❝❛t✐♦♥ ✐s ✐♠♣♦rt❛♥t ❢♦r ♠❛t❤❡♠❛t✐❝❛❧ s♦❧✉t✐♦♥ t❡❝❤♥✐q✉❡✱ ❜✉t ♥♦t ❢♦r s✐♠✉❧❛t✐♦♥ ✐♥ ❛ ♣r♦❣r❛♠✳

❋✐❜♦♥❛❝❝✐ ❞❡r✐✈❡❞ t❤❡ s❡q✉❡♥❝❡ ❜② ♠♦❞❡❧✐♥❣ r❛t ♣♦♣✉❧❛t✐♦♥s✱ ❜✉t t❤❡ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ❤❛s ❛ r❛♥❣❡ ♦❢ ♣❡❝✉❧✐❛r ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❤❛s t❤❡r❡❢♦r❡ ❛ttr❛❝t❡❞ ♠✉❝❤ ❛tt❡♥t✐♦♥ ❢r♦♠ ♠❛t❤❡♠❛t✐❝✐❛♥s✳

Pr♦❣r❛♠ ❢♦r ❣❡♥❡r❛t✐♥❣ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs

◆ ❂ ✐♥t✭s②s✳❛r❣✈❬✶❪✮ ❢r♦♠ ♥✉♠♣② ✐♠♣♦rt ③❡r♦s ① ❂ ③❡r♦s✭◆✰✶✱ ✐♥t✮ ①❬✵❪ ❂ ✶ ①❬✶❪ ❂ ✶ ❢♦r ♥ ✐♥ r❛♥❣❡✭✷✱ ◆✰✶✮✿ ①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ①❬♥✲✷❪ ♣r✐♥t ♥✱ ①❬♥❪

❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs ❝❛♥ ❝❛✉s❡ ♦✈❡r✢♦✇ ✐♥ ◆✉♠P② ❛rr❛②s

❘✉♥ t❤❡ ♣r♦❣r❛♠ ✇✐t❤ N = ✺✵✿

✷ ✷ ✸ ✸ ✹ ✺ ✺ ✽ ✻ ✶✸ ✳✳✳ ✹✺ ✶✽✸✻✸✶✶✾✵✸ ❲❛r♥✐♥❣✿ ♦✈❡r❢❧♦✇ ❡♥❝♦✉♥t❡r❡❞ ✐♥ ❧♦♥❣❴s❝❛❧❛rs ✹✻ ✲✶✸✷✸✼✺✷✷✷✸

◆♦t❡✿ ❈❤❛♥❣✐♥❣ ✐♥t t♦ ❧♦♥❣ ♦r ✐♥t✻✹ ❢♦r ❛rr❛② ❡❧❡♠❡♥ts ❛❧❧♦✇s N ≤ ✾✶ ❈❛♥ ✉s❡ ❢❧♦❛t✾✻ ✭t❤♦✉❣❤ xn ✐s ✐♥t❡❣❡r✮✿ N ≤ ✷✸✻✵✵

◆♦ ♦✈❡r✢♦✇ ✇❤❡♥ ✉s✐♥❣ P②t❤♦♥ ✐♥t t②♣❡s

❇❡st✿ ✉s❡ P②t❤♦♥ s❝❛❧❛rs ♦❢ t②♣❡ ✐♥t ✲ t❤❡s❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❤❛♥❣❡s t♦ ❧♦♥❣ ✇❤❡♥ ♦✈❡r✢♦✇ ✐♥ ✐♥t ❚❤❡ ❧♦♥❣ t②♣❡ ✐♥ P②t❤♦♥ ❤❛s ❛r❜✐tr❛r✐❧② ♠❛♥② ❞✐❣✐ts ✭❛s ♠❛♥② ❛s r❡q✉✐r❡❞ ✐♥ ❛ ❝♦♠♣✉t❛t✐♦♥✦✮ ◆♦t❡✿ ❧♦♥❣ ❢♦r ❛rr❛②s ✐s ✻✹✲❜✐t ✐♥t❡❣❡r ✭✐♥t✻✹✮✱ ✇❤✐❧❡ s❝❛❧❛r ❧♦♥❣ ✐♥ P②t❤♦♥ ✐s ❛♥ ✐♥t❡❣❡r ✇✐t❤ ❛s ✏✐♥✜♥✐t❡❧②✑ ♠❛♥② ❞✐❣✐ts

Pr♦❣r❛♠ ✇✐t❤ P②t❤♦♥✬s ✐♥t t②♣❡ ❢♦r ✐♥t❡❣❡rs

❚❤❡ ♣r♦❣r❛♠ ♥♦✇ ❛✈♦✐❞s ❛rr❛②s ❛♥❞ ♠❛❦❡s ✉s❡ ♦❢ t❤r❡❡ ✐♥t ♦❜❥❡❝ts ✭✇❤✐❝❤ ❛✉t♦♠❛t✐❝❛❧❧② ❝❤❛♥❣❡s t♦ ❧♦♥❣ ✇❤❡♥ ♥❡❡❞❡❞✮✿

✐♠♣♦rt s②s ◆ ❂ ✐♥t✭s②s✳❛r❣✈❬✶❪✮ ①♥♠✶ ❂ ✶ ★ ✧①❴♥ ♠✐♥✉s ✶✧ ①♥♠✷ ❂ ✶ ★ ✧①❴♥ ♠✐♥✉s ✷✧ ♥ ❂ ✷ ✇❤✐❧❡ ♥ ❁❂ ◆✿ ①♥ ❂ ①♥♠✶ ✰ ①♥♠✷ ♣r✐♥t ✬①❴✪❞ ❂ ✪❞✬ ✪ ✭♥✱ ①♥✮ ①♥♠✷ ❂ ①♥♠✶ ①♥♠✶ ❂ ①♥ ♥ ✰❂ ✶

❘✉♥ ✇✐t❤ N = ✷✵✵✿

①❴✷ ❂ ✷ ①❴✸ ❂ ✸ ✳✳✳ ①❴✶✾✽ ❂ ✶✼✸✹✵✷✺✷✶✶✼✷✼✾✼✽✶✸✶✺✾✻✽✺✵✸✼✷✽✹✸✼✶✾✹✷✵✹✹✸✵✶ ①❴✶✾✾ ❂ ✷✽✵✺✼✶✶✼✷✾✾✷✺✶✵✶✹✵✵✸✼✻✶✶✾✸✷✹✶✸✵✸✽✻✼✼✶✽✾✺✷✺ ①❴✷✵✵ ❂ ✹✺✸✾✼✸✻✾✹✶✻✺✸✵✼✾✺✸✶✾✼✷✾✻✾✻✾✻✾✼✹✶✵✻✶✾✷✸✸✽✷✻

▲✐♠✐t✐♦♥✿ ②♦✉r ❝♦♠♣✉t❡r✬s ♠❡♠♦r②

◆❡✇ ♣r♦❜❧❡♠ s❡tt✐♥❣✿ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✇✐t❤ ❧✐♠✐t❡❞ ❡♥✈✐r♦♥♠❡♥t❛❧ r❡s♦✉r❝❡s

❚❤❡ ♠♦❞❡❧ ❢♦r ❣r♦✇t❤ ♦❢ ♠♦♥❡② ✐♥ ❛ ❜❛♥❦ ❤❛s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ t②♣❡ xn = x✵C n (= x✵en ❧♥ C) ◆♦t❡✿ ❚❤✐s ✐s ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ✐♥ t✐♠❡ ✭n✮ P♦♣✉❧❛t✐♦♥s ♦❢ ❤✉♠❛♥s✱ ❛♥✐♠❛❧s✱ ❛♥❞ ❝❡❧❧s ❛❧s♦ ❡①❤✐❜✐t t❤❡ s❛♠❡ t②♣❡ ♦❢ ❣r♦✇t❤ ❛s ❧♦♥❣ ❛s t❤❡r❡ ❛r❡ ✉♥❧✐♠✐t❡❞ r❡s♦✉r❝❡s ✭s♣❛❝❡ ❛♥❞ ❢♦♦❞✮ ▼♦st ❡♥✈✐r♦♥♠❡♥ts ❝❛♥ ♦♥❧② s✉♣♣♦rt ❛ ♠❛①✐♠✉♠ ♥✉♠❜❡r M ♦❢ ✐♥❞✐✈✐❞✉❛❧s ❍♦✇ ❝❛♥ ✇❡ ♠♦❞❡❧ t❤✐s ❧✐♠✐t❛t✐♦♥❄

▼♦❞❡❧✐♥❣ ❣r♦✇t❤ ✐♥ ❛♥ ❡♥✈✐r♦♥♠❡♥t ✇✐t❤ ❧✐♠✐t❡❞ r❡s♦✉r❝❡s

■♥✐t✐❛❧❧②✱ ✇❤❡♥ t❤❡r❡ ❛r❡ ❡♥♦✉❣❤ r❡s♦✉r❝❡s✱ t❤❡ ❣r♦✇t❤ ✐s ❡①♣♦♥❡♥t✐❛❧✿ xn = xn−✶ + r ✶✵✵xn−✶ ❚❤❡ ❣r♦✇t❤ r❛t❡ r ♠✉st ❞❡❝❛② t♦ ③❡r♦ ❛s xn ❛♣♣r♦❛❝❤❡s M✳ ❚❤❡ s✐♠♣❧❡st ✈❛r✐❛t✐♦♥ ♦❢ r(n) ✐s ❛ ❧✐♥❡❛r✿ r(n) = ̺

• ✶ − xn

M

• ❖❜s❡r✈❡✿ r(n) ≈ ̺ ❢♦r s♠❛❧❧ n ✇❤❡♥ xn ≪ M✱ ❛♥❞ r(n) → ✵ ❛s

xn → M ❛♥❞ n ✐s ❜✐❣ ▲♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧✿ xn = xn−✶ + ̺ ✶✵✵xn−✶

• ✶ − xn−✶

M

• ✭❚❤✐s ✐s ❛ ♥♦♥❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✮

❚❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❧♦❣✐st✐❝ ❣r♦✇t❤

■♥ ❛ ♣r♦❣r❛♠ ✐t ✐s ❡❛s② t♦ ✐♥tr♦❞✉❝❡ ❧♦❣✐st✐❝ ✐♥st❡❛❞ ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤✱ ❥✉st r❡♣❧❛❝❡

①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ♣✴✶✵✵✳✵✮✯①❬♥✲✶❪

❜②

①❬♥❪ ❂ ①❬♥✲✶❪ ✰ ✭r❤♦✴✶✵✵✳✵✮✯①❬♥✲✶❪✯✭✶ ✲ ①❬♥✲✶❪✴❢❧♦❛t✭▼✮✮

100 150 200 250 300 350 400 450 500 50 100 150 200 no of individuals time units

❚❤❡ ❢❛❝t♦r✐❛❧ ❛s ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

❚❤❡ ❢❛❝t♦r✐❛❧ n! ✐s ❞❡✜♥❡❞ ❛s n(n − ✶)(n − ✷) · · · ✶, ✵! = ✶ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❤❛s xn = n! ❛s s♦❧✉t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❢❛❝t♦r✐❛❧✿ xn = nxn−✶, x✵ = ✶

❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ♠✉st ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥

■♥ ♠❛t❤❡♠❛t✐❝s✱ ✐t ✐s ♠✉❝❤ str❡ss❡❞ t❤❛t ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❢♦r xn ♠✉st ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ x✵ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ♦❜✈✐♦✉s ✇❤❡♥ ♣r♦❣r❛♠♠✐♥❣✿ ♦t❤❡r✇✐s❡ ✇❡ ❝❛♥♥♦t st❛rt t❤❡ ♣r♦❣r❛♠ ✭x✵ ✐s ♥❡❡❞❡❞ t♦ ❝♦♠♣✉t❡ xn✮ ❍♦✇❡✈❡r✿ ✐❢ ②♦✉ ❢♦r❣❡t ①❬✵❪ ❂ ①✵ ✐♥ t❤❡ ♣r♦❣r❛♠✱ ②♦✉ ❣❡t x✵ = ✵ ✭❜❡❝❛✉s❡ ① ❂ ③❡r♦❡s✭◆✰✶✮✮✱ ✇❤✐❝❤ ✭✉s✉❛❧❧②✮ ❣✐✈❡s ✉♥✐♥t❡♥❞❡❞ r❡s✉❧ts✦

❍❛✈❡ ②♦✉ ❡✈❡r t❤♦✉❣❤ ❛❜♦✉t ❤♦✇ s✐♥ x ✐s r❡❛❧❧② ❝❛❧❝✉❧❛t❡❞❄

❍♦✇ ❝❛♥ ②♦✉ ❝❛❧❝✉❧❛t❡ s✐♥ x✱ ❧♥ x✱ ex ✇✐t❤♦✉t ❛ ❝❛❧❝✉❧❛t♦r ♦r ♣r♦❣r❛♠❄ ❚❤❡s❡ ❢✉♥❝t✐♦♥s ✇❡r❡ ♦r✐❣✐♥❛❧❧② ❞❡✜♥❡❞ t♦ ❤❛✈❡ s♦♠❡ ❞❡s✐r❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦♣❡rt✐❡s✱ ❜✉t ✇✐t❤♦✉t ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❤♦✇ t♦ ❡✈❛❧✉❛t❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡s ■❞❡❛✿ ❛♣♣r♦①✐♠❛t❡ s✐♥ x✱ ❡t❝✳ ❜② ♣♦❧②♥♦♠✐❛❧s✱ s✐♥❝❡ t❤❡② ❛r❡ ❡❛s② t♦ ❝❛❧❝✉❧❛t❡ ✭s✉♠✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✮✱ ❜✉t ❤♦✇❄❄

❲♦✉❧❞ ②♦✉ ❡①♣❡❝t t❤❡s❡ ❢❛♥t❛st✐❝ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts❄

❆♠❛③✐♥❣ r❡s✉❧t ❜② ●r❡❣♦r②✱ ✶✻✻✼✿ s✐♥ x =

∞

• k=✵

(−✶)k x✷k+✶ (✷k + ✶)! ❊✈❡♥ ♠♦r❡ ❛♠❛③✐♥❣ r❡s✉❧t ❜② ❚❛②❧♦r✱ ✶✼✶✺✿ f (x) =

∞

• k=✵

✶ k!( dk dxk f (✵))xk ❋♦r ✏❛♥②✑ f (x)✱ ✐❢ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡✱ ❛❞❞✱ ❛♥❞ ♠✉❧t✐♣❧② xk✱ ✇❡ ❝❛♥ ❡✈❛❧✉❛t❡ f ❛t ❛♥② x ✭✦✦✦✮

❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s

Pr❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ✇♦r❦s ✇✐t❤ ❛ tr✉♥❝❛t❡❞ s✉♠✿ f (x) ≈

N

• k=✵

✶ k!( dk dxk f (✵))xk N = ✶ ✐s ✈❡r② ♣♦♣✉❧❛r ❛♥❞ ❤❛s ❜❡❡♥ ❡ss❡♥t✐❛❧ ✐♥ ❞❡✈❡❧♦♣✐♥❣ ♣❤②s✐❝s ❛♥❞ t❡❝❤♥♦❧♦❣② ❊①❛♠♣❧❡✿ ex =

∞

• k=✵

xk k! ≈ ✶ + x + ✶ ✷x✷ + ✶ ✻x✸ ≈ ✶ + x

❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❛r♦✉♥❞ ❛♥ ❛r❜✐tr❛r② ♣♦✐♥t

❚❤❡ ♣r❡✈✐♦✉s ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ♠♦st ❛❝❝✉r❛t❡ ❛r♦✉♥❞ x = ✵✳ ❈❛♥ ♠❛❦❡ t❤❡ ♣♦❧②♥♦♠✐❛❧s ❛❝❝✉r❛t❡ ❛r♦✉♥❞ ❛♥② ♣♦✐♥t x = a✿ f (x) ≈

N

• k=✵

✶ k!( dk dxk f (a))(x − a)k

❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ❛s ♦♥❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

❚❤❡ ❚❛②❧♦r s❡r✐❡s ❢♦r ex ❛r♦✉♥❞ x = ✵ r❡❛❞s ex =

∞

• n=✵

xn n! ❉❡✜♥❡ en =

n−✶

• k=✵

xk k! =

n−✷

• k=✵

xk k! + xn−✶ (n − ✶)! ❲❡ ❝❛♥ ❢♦r♠✉❧❛t❡ t❤❡ s✉♠ ✐♥ en ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ en = en−✶ + xn−✶ (n − ✶)!, e✵ = ✵

▼♦r❡ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥✿ t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ❛s t✇♦ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s

❖❜s❡r✈❡✿ xn n! = xn−✶ (n − ✶)! · x n ▲❡t an = xn/n!✳ ❚❤❡♥ ✇❡ ❝❛♥ ❡✣❝✐❡♥t❧② ❝♦♠♣✉t❡ an ✈✐❛ an = an−✶ x n, a✵ = ✶ ◆♦✇ ✇❡ ❝❛♥ ✉♣❞❛t❡ ❡❛❝❤ t❡r♠ ✈✐❛ t❤❡ an ❡q✉❛t✐♦♥ ❛♥❞ s✉♠ t❤❡ t❡r♠s ✈✐❛ t❤❡ en ❡q✉❛t✐♦♥✿ en = en−✶ + an−✶, e✵ = ✵, a✵ = ✶ an = x nan−✶ ❙❡❡ t❤❡ ❜♦♦❦ ❢♦r ♠♦r❡ ❞❡t❛✐❧s

◆♦♥❧✐♥❡❛r ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s

• ❡♥❡r✐❝ ❢♦r♠ ♦❢ ❛♥② ✭❛❧❣❡❜r❛✐❝✮ ❡q✉❛t✐♦♥ ✐♥ x✿

f (x) = ✵ ❊①❛♠♣❧❡s t❤❛t ❝❛♥ ❜❡ s♦❧✈❡❞ ❜② ❤❛♥❞✿ ax + b = ✵ ax✷ + bx + c = ✵ s✐♥ x + ❝♦s x = ✶ ❙✐♠♣❧❡ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❝❛♥ s♦❧✈❡ ✏❛♥②✑ ❡q✉❛t✐♦♥ f (x) = ✵ ❙❛❢❡st✿ ❇✐s❡❝t✐♦♥ ❋❛st❡st✿ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ❉♦♥✬t ❧✐❦❡ f ′(x) ✐♥ ◆❡✇t♦♥✬s ♠❡t❤♦❞❄ ❯s❡ t❤❡ ❙❡❝❛♥t ♠❡t❤♦❞ ❙❡❝❛♥t ❛♥❞ ◆❡✇t♦♥ ❛r❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✦

◆❡✇t♦♥✬s ♠❡t❤♦❞ ❢♦r ✜♥❞✐♥❣ ③❡r♦s❀ ✐❧❧✉str❛t✐♦♥

◆❡✇t♦♥✬s ♠❡t❤♦❞ ❢♦r ✜♥❞✐♥❣ ③❡r♦s❀ ♠❛t❤❡♠❛t✐❝s

◆❡✇t♦♥✬s ♠❡t❤♦❞ ❙✐♠♣s♦♥ ✭✶✼✹✵✮ ❝❛♠❡ ✉♣ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ f (x) = ✵ ✭❜❛s❡❞ ♦♥ ✐❞❡❛s ❜② ◆❡✇t♦♥✮✿ xn = xn−✶ − f (xn−✶) f ′(xn−✶), x✵ ❣✐✈❡♥ ◆♦t❡✿ ❚❤✐s ✐s ❛ ✭♥♦♥❧✐♥❡❛r✦✮ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ❆s n → ∞✱ ✇❡ ❤♦♣❡ t❤❛t xn → xs✱ ✇❤❡r❡ xs s♦❧✈❡s f (xs) = ✵ ❍♦✇ t♦ ❝❤♦♦s❡ N ✇❤❡♥ ✇❤❛t ✇❡ ✇❛♥t ✐s xN ❝❧♦s❡ t♦ xs❄ ◆❡❡❞ ❛ s❧✐❣❤t❧② ❞✐✛❡r❡♥t ♣r♦❣r❛♠✿ s✐♠✉❧❛t❡ ✉♥t✐❧ f (x) ≤ ǫ✱ ✇❤❡r❡ ǫ ✐s ❛ s♠❛❧❧ t♦❧❡r❛♥❝❡ ❈❛✉t✐♦♥✿ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ♠❛② ✭❡❛s✐❧②✮ ❞✐✈❡r❣❡✱ s♦ f (x) ≤ ǫ ♠❛② ♥❡✈❡r ♦❝❝✉r✦

❆ ♣r♦❣r❛♠ ❢♦r ◆❡✇t♦♥✬s ♠❡t❤♦❞

◗✉✐❝❦ ✐♠♣❧❡♠❡♥t❛t✐♦♥✿

❞❡❢ ◆❡✇t♦♥✭❢✱ ①✱ ❞❢❞①✱ ❡♣s✐❧♦♥❂✶✳✵❊✲✼✱ ♠❛①❴♥❂✶✵✵✮✿ ♥ ❂ ✵ ✇❤✐❧❡ ❛❜s✭❢✭①✮✮ ❃ ❡♣s✐❧♦♥ ❛♥❞ ♥ ❁❂ ♠❛①❴♥✿ ① ❂ ① ✲ ❢✭①✮✴❞❢❞①✭①✮ ♥ ✰❂ ✶ r❡t✉r♥ ①✱ ♥✱ ❢✭①✮

◆♦t❡✿ f (x) ✐s ❡✈❛❧✉❛t❡❞ t✇✐❝❡ ✐♥ ❡❛❝❤ ♣❛ss ♦❢ t❤❡ ❧♦♦♣ ✲ ♦♥❧② ♦♥❡ ❡✈❛❧✉❛t✐♦♥ ✐s str✐❝t❧② ♥❡❝❡ss❛r② ✭❝❛♥ st♦r❡ t❤❡ ✈❛❧✉❡ ✐♥ ❛ ✈❛r✐❛❜❧❡ ❛♥❞ r❡✉s❡ ✐t✮ ❢✭①✮✴❞❢❞①✭①✮ ❝❛♥ ❣✐✈❡ ✐♥t❡❣❡r ❞✐✈✐s✐♦♥ ■t ❝♦✉❧❞ ❜❡ ❤❛♥❞② t♦ st♦r❡ t❤❡ ① ❛♥❞ ❢✭①✮ ✈❛❧✉❡s ✐♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ ✭❢♦r ♣❧♦tt✐♥❣ ♦r ♣r✐♥t✐♥❣ ❛ ❝♦♥✈❡r❣❡♥❝❡ t❛❜❧❡✮

❆♥ ✐♠♣r♦✈❡❞ ❢✉♥❝t✐♦♥ ❢♦r ◆❡✇t♦♥✬s ♠❡t❤♦❞

❖♥❧② ♦♥❡ f (x) ❝❛❧❧ ✐♥ ❡❛❝❤ ✐t❡r❛t✐♦♥✱ ♦♣t✐♦♥❛❧ st♦r❛❣❡ ♦❢ (x, f (x)) ✈❛❧✉❡s ❞✉r✐♥❣ t❤❡ ✐t❡r❛t✐♦♥s✱ ❛♥❞ ❡♥s✉r❡❞ ✢♦❛t ❞✐✈✐s✐♦♥✿

❞❡❢ ◆❡✇t♦♥✭❢✱ ①✱ ❞❢❞①✱ ❡♣s✐❧♦♥❂✶✳✵❊✲✼✱ ♠❛①❴♥❂✶✵✵✱ st♦r❡❂❋❛❧s❡✮✿ ❢❴✈❛❧✉❡ ❂ ❢✭①✮ ♥ ❂ ✵ ✐❢ st♦r❡✿ ✐♥❢♦ ❂ ❬✭①✱ ❢❴✈❛❧✉❡✮❪ ✇❤✐❧❡ ❛❜s✭❢❴✈❛❧✉❡✮ ❃ ❡♣s✐❧♦♥ ❛♥❞ ♥ ❁❂ ♠❛①❴♥✿ ① ❂ ① ✲ ❢❧♦❛t✭❢❴✈❛❧✉❡✮✴❞❢❞①✭①✮ ♥ ✰❂ ✶ ❢❴✈❛❧✉❡ ❂ ❢✭①✮ ✐❢ st♦r❡✿ ✐♥❢♦✳❛♣♣❡♥❞✭✭①✱ ❢❴✈❛❧✉❡✮✮ ✐❢ st♦r❡✿ r❡t✉r♥ ①✱ ✐♥❢♦ ❡❧s❡✿ r❡t✉r♥ ①✱ ♥✱ ❢❴✈❛❧✉❡

❆♣♣❧✐❝❛t✐♦♥ ♦❢ ◆❡✇t♦♥✬s ♠❡t❤♦❞

e−✵.✶x✷ s✐♥(π ✷ x) = ✵ ❙♦❧✉t✐♦♥s✿ x = ✵, ±✷, ±✹, ±✻, . . . ▼❛✐♥ ♣r♦❣r❛♠✿

❢r♦♠ ♠❛t❤ ✐♠♣♦rt s✐♥✱ ❝♦s✱ ❡①♣✱ ♣✐ ✐♠♣♦rt s②s ❞❡❢ ❣✭①✮✿ r❡t✉r♥ ❡①♣✭✲✵✳✶✯①✯✯✷✮✯s✐♥✭♣✐✴✷✯①✮ ❞❡❢ ❞❣✭①✮✿ r❡t✉r♥ ✲✷✯✵✳✶✯①✯❡①♣✭✲✵✳✶✯①✯✯✷✮✯s✐♥✭♣✐✴✷✯①✮ ✰ ❭ ♣✐✴✷✯❡①♣✭✲✵✳✶✯①✯✯✷✮✯❝♦s✭♣✐✴✷✯①✮ ①✵ ❂ ❢❧♦❛t✭s②s✳❛r❣✈❬✶❪✮ ①✱ ✐♥❢♦ ❂ ◆❡✇t♦♥✭❣✱ ①✵✱ ❞❣✱ st♦r❡❂❚r✉❡✮ ♣r✐♥t ✬❈♦♠♣✉t❡❞ ③❡r♦✿✬✱ ① ★ Pr✐♥t t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥ ★ ✭✐✳❡✳✱ t❤❡ s❡❛r❝❤ ❢♦r t❤❡ r♦♦t✮ ❢♦r ✐ ✐♥ r❛♥❣❡✭❧❡♥✭✐♥❢♦✮✮✿ ♣r✐♥t ✬■t❡r❛t✐♦♥ ✪✸❞✿ ❢✭✪❣✮❂✪❣✬ ✪ ✭✐✱ ✐♥❢♦❬✐❪❬✵❪✱ ✐♥❢♦❬✐❪❬✶❪✮

❘❡s✉❧ts ❢r♦♠ t❤✐s t❡st ♣r♦❜❧❡♠

x✵ = ✶.✼ ❣✐✈❡s q✉✐❝❦ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s t❤❡ ❝❧♦s❡st r♦♦t x = ✵✿

③❡r♦✿ ✶✳✾✾✾✾✾✾✾✾✾✼✻✽✹✹✾ ■t❡r❛t✐♦♥ ✵✿ ❢✭✶✳✼✮❂✵✳✸✹✵✵✹✹ ■t❡r❛t✐♦♥ ✶✿ ❢✭✶✳✾✾✷✶✺✮❂✵✳✵✵✽✷✽✼✽✻ ■t❡r❛t✐♦♥ ✷✿ ❢✭✶✳✾✾✾✾✽✮❂✷✳✺✸✸✹✼❡✲✵✺ ■t❡r❛t✐♦♥ ✸✿ ❢✭✷✮❂✷✳✹✸✽✵✽❡✲✶✵

❙t❛rt ✈❛❧✉❡ x✵ = ✸ ✭❝❧♦s❡st r♦♦t x = ✷ ♦r x = ✹✮✿

③❡r♦✿ ✹✷✳✹✾✼✷✸✸✶✻✵✶✶✸✻✷ ■t❡r❛t✐♦♥ ✵✿ ❢✭✸✮❂✲✵✳✹✵✻✺✼ ■t❡r❛t✐♦♥ ✶✿ ❢✭✹✳✻✻✻✻✼✮❂✵✳✵✾✽✶✶✹✻ ■t❡r❛t✐♦♥ ✷✿ ❢✭✹✷✳✹✾✼✷✮❂✲✷✳✺✾✵✸✼❡✲✼✾

❲❤❛t ❤❛♣♣❡♥❡❞ ❤❡r❡❄❄

❚r② t❤❡ ❞❡♠♦ ♣r♦❣r❛♠ sr❝✴❞✐❢❢❡q✴◆❡✇t♦♥❴♠♦✈✐❡✳♣② ✇✐t❤ x✵ = ✸✱ x ∈ [−✷, ✺✵] ❢♦r ♣❧♦tt✐♥❣ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ′(x)✿

❚❡r♠✐♥❛❧❃ ♣②t❤♦♥ ◆❡✇t♦♥❴♠♦✈✐❡✳♣② ✧❡①♣✭✲✵✳✶✯①✯✯✷✮✯s✐♥✭♣✐✴✷✯①✮✧ ❭ ♥✉♠❡r✐❝ ✸ ✲✷ ✺✵

▲❡ss♦♥ ❧❡❛r♥❡❞✿ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ♠❛② ✇♦r❦ ✜♥❡ ♦r ❣✐✈❡ ✇r♦♥❣ r❡s✉❧ts✦ ❨♦✉ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❡t❤♦❞ t♦ ✐♥t❡r♣r❡t t❤❡ r❡s✉❧ts✦

❋✐rst st❡♣✿ ✇❡✬r❡ ♠♦✈✐♥❣ t♦ t❤❡ r✐❣❤t ✭x = ✹❄✮

10 20 30 40 50 0.5 0.0 0.5

approximate root = 3; f(3) = -0.40657 f(x)

• approx. root

y=0

• approx. line

❙❡❝♦♥❞ st❡♣✿ ♦♦♣s✱ t♦♦ ♠✉❝❤ t♦ t❤❡ r✐❣❤t✳✳✳

10 20 30 40 50 0.5 0.0 0.5

approximate root = 4.66667; f(4.66667) = 0.0981146 f(x)

• approx. root

y=0

• approx. line

❚❤✐r❞ st❡♣✿ ❞✐s❛st❡r s✐♥❝❡ ✇❡✬r❡ ✏❞♦♥❡✑ ✭f (x) ≈ ✵✮

10 20 30 40 50 0.5 0.0 0.5

approximate root = 42.4972; f(42.4972) = -2.5906e-79 f(x)

• approx. root

y=0

• approx. line

Pr♦❣r❛♠♠✐♥❣ ✇✐t❤ s♦✉♥❞

❚♦♥❡s ❛r❡ s✐♥❡ ✇❛✈❡s✿ ❆ t♦♥❡ ❆ ✭✹✹✵ ❍③✮ ✐s ❛ s✐♥❡ ✇❛✈❡ ✇✐t❤ ❢r❡q✉❡♥❝② ✹✹✵ ❍③✿ s(t) = A s✐♥ (✷πft) , f = ✹✹✵ ❖♥ ❛ ❝♦♠♣✉t❡r ✇❡ r❡♣r❡s❡♥t s(t) ❜② ❛ ❞✐s❝r❡t❡ s❡t ♦❢ ♣♦✐♥ts ♦♥ t❤❡ ❢✉♥❝t✐♦♥ ❝✉r✈❡ ✭❡①❛❝t❧② ❛s ✇❡ ❞♦ ✇❤❡♥ ✇❡ ♣❧♦t s(t)✮✳ ❈❉ q✉❛❧✐t② ♥❡❡❞s ✹✹✶✵✵ s❛♠♣❧❡s ♣❡r s❡❝♦♥❞✳

▼❛❦✐♥❣ ❛ s♦✉♥❞ ✜❧❡ ✇✐t❤ s✐♥❣❧❡ t♦♥❡ ✭♣❛rt ✶✮

r✿ s❛♠♣❧✐♥❣ r❛t❡ ✭s❛♠♣❧❡s ♣❡r s❡❝♦♥❞✱ ❞❡❢❛✉❧t ✹✹✶✵✵✮ f ✿ ❢r❡q✉❡♥❝② ♦❢ t❤❡ t♦♥❡ m✿ ❞✉r❛t✐♦♥ ♦❢ t❤❡ t♦♥❡ ✭s❡❝♦♥❞s✮ ❙❛♠♣❧❡❞ s✐♥❡ ❢✉♥❝t✐♦♥ ❢♦r t❤✐s t♦♥❡✿ sn = A s✐♥

• ✷πf n

r

• ,

n = ✵, ✶, . . . , m · r ❈♦❞❡ ✭✇❡ ✉s❡ ❞❡s❝r✐♣t✐✈❡ ♥❛♠❡s✿ ❢r❡q✉❡♥❝② f ✱ ❧❡♥❣t❤ m✱ ❛♠♣❧✐t✉❞❡ A✱ s❛♠♣❧❡❴r❛t❡ r✮✿

✐♠♣♦rt ♥✉♠♣② ❞❡❢ ♥♦t❡✭❢r❡q✉❡♥❝②✱ ❧❡♥❣t❤✱ ❛♠♣❧✐t✉❞❡❂✶✱ s❛♠♣❧❡❴r❛t❡❂✹✹✶✵✵✮✿ t✐♠❡❴♣♦✐♥ts ❂ ♥✉♠♣②✳❧✐♥s♣❛❝❡✭✵✱ ❧❡♥❣t❤✱ ❧❡♥❣t❤✯s❛♠♣❧❡❴r❛t❡✮ ❞❛t❛ ❂ ♥✉♠♣②✳s✐♥✭✷✯♥✉♠♣②✳♣✐✯❢r❡q✉❡♥❝②✯t✐♠❡❴♣♦✐♥ts✮ ❞❛t❛ ❂ ❛♠♣❧✐t✉❞❡✯❞❛t❛ r❡t✉r♥ ❞❛t❛

▼❛❦✐♥❣ ❛ s♦✉♥❞ ✜❧❡ ✇✐t❤ s✐♥❣❧❡ t♦♥❡ ✭♣❛rt ✷✮

❲❡ ❤❛✈❡ ❞❛t❛ ❛s ❛♥ ❛rr❛② ✇✐t❤ ❢❧♦❛t ❛♥❞ ✉♥✐t ❛♠♣❧✐t✉❞❡ ❙♦✉♥❞ ❞❛t❛ ✐♥ ❛ ✜❧❡ s❤♦✉❧❞ ❤❛✈❡ ✷✲❜②t❡ ✐♥t❡❣❡rs ✭✐♥t✶✻✮ ❛s ❞❛t❛ ❡❧❡♠❡♥ts ❛♥❞ ❛♠♣❧✐t✉❞❡s ✉♣ t♦ ✷✶✺ − ✶ ✭♠❛① ✈❛❧✉❡ ❢♦r ✐♥t✶✻ ❞❛t❛✮

❞❛t❛ ❂ ♥♦t❡✭✹✹✵✱ ✷✮ ❞❛t❛ ❂ ❞❛t❛✳❛st②♣❡✭♥✉♠♣②✳✐♥t✶✻✮ ♠❛①❴❛♠♣❧✐t✉❞❡ ❂ ✷✯✯✶✺ ✲ ✶ ❞❛t❛ ❂ ♠❛①❴❛♠♣❧✐t✉❞❡✯❞❛t❛ ✐♠♣♦rt s❝✐t♦♦❧s✳s♦✉♥❞ s❝✐t♦♦❧s✳s♦✉♥❞✳✇r✐t❡✭❞❛t❛✱ ✬❆t♦♥❡✳✇❛✈✬✮ s❝✐t♦♦❧s✳s♦✉♥❞✳♣❧❛②✭✬❆t♦♥❡✳✇❛✈✬✮

❘❡❛❞✐♥❣ s♦✉♥❞ ❢r♦♠ ✜❧❡

▲❡t ✉s r❡❛❞ ❛ s♦✉♥❞ ✜❧❡ ❛♥❞ ❛❞❞ ❡❝❤♦ ❙♦✉♥❞ ❂ ❛rr❛② s❬♥❪ ❊❝❤♦ ♠❡❛♥s t♦ ❛❞❞ ❛ ❞❡❧❛② ♦❢ t❤❡ s♦✉♥❞

★ ❡❝❤♦✿ ❡❬♥❪ ❂ ❜❡t❛✯s❬♥❪ ✰ ✭✶✲❜❡t❛✮✯s❬♥✲❜❪ ❞❡❢ ❛❞❞❴❡❝❤♦✭❞❛t❛✱ ❜❡t❛❂✵✳✽✱ ❞❡❧❛②❂✵✳✵✵✷✱ s❛♠♣❧❡❴r❛t❡❂✹✹✶✵✵✮✿ ♥❡✇❞❛t❛ ❂ ❞❛t❛✳❝♦♣②✭✮ s❤✐❢t ❂ ✐♥t✭❞❡❧❛②✯s❛♠♣❧❡❴r❛t❡✮ ★ ❜ ✭♠❛t❤ s②♠❜♦❧✮ ❢♦r ✐ ✐♥ ①r❛♥❣❡✭s❤✐❢t✱ ❧❡♥✭❞❛t❛✮✮✿ ♥❡✇❞❛t❛❬✐❪ ❂ ❜❡t❛✯❞❛t❛❬✐❪ ✰ ✭✶✲❜❡t❛✮✯❞❛t❛❬✐✲s❤✐❢t❪ r❡t✉r♥ ♥❡✇❞❛t❛

▲♦❛❞ ❞❛t❛✱ ❛❞❞ ❡❝❤♦ ❛♥❞ ♣❧❛②✿

❞❛t❛ ❂ s❝✐t♦♦❧s✳s♦✉♥❞✳r❡❛❞✭❢✐❧❡♥❛♠❡✮ ❞❛t❛ ❂ ❞❛t❛✳❛st②♣❡✭❢❧♦❛t✮ ❞❛t❛ ❂ ❛❞❞❴❡❝❤♦✭❞❛t❛✱ ❜❡t❛❂✵✳✻✮ ❞❛t❛ ❂ ❞❛t❛✳❛st②♣❡✭✐♥t✶✻✮ s❝✐t♦♦❧s✳s♦✉♥❞✳♣❧❛②✭❞❛t❛✮

P❧❛②✐♥❣ ♠❛♥② ♥♦t❡s

❊❛❝❤ ♥♦t❡ ✐s ❛♥ ❛rr❛② ♦❢ s❛♠♣❧❡s ❢r♦♠ ❛ s✐♥❡ ✇✐t❤ ❛ ❢r❡q✉❡♥❝② ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ♥♦t❡ ❆ss✉♠❡ ✇❡ ❤❛✈❡ s❡✈❡r❛❧ ♥♦t❡ ❛rr❛②s ❞❛t❛✶✱ ❞❛t❛✷✱ ✳✳✳✿

★ ♣✉t ❞❛t❛✶✱ ❞❛t❛✷✱ ✳✳✳ ❛❢t❡r ❡❛❝❤ ♦t❤❡r ✐♥ ❛ ♥❡✇ ❛rr❛②✿ ❞❛t❛ ❂ ♥✉♠♣②✳❝♦♥❝❛t❡♥❛t❡✭✭❞❛t❛✶✱ ❞❛t❛✷✱ ❞❛t❛✸✱ ✳✳✳✮✮

❚❤❡ st❛rt ♦❢ ✧◆♦t❤✐♥❣ ❊❧s❡ ▼❛tt❡rs✧ ✭▼❡t❛❧❧✐❝❛✮✿

❊✶ ❂ ♥♦t❡✭✶✻✹✳✽✶✱ ✳✺✮

• ❂ ♥♦t❡✭✸✾✷✱ ✳✺✮

❇ ❂ ♥♦t❡✭✹✾✸✳✽✽✱ ✳✺✮ ❊✷ ❂ ♥♦t❡✭✻✺✾✳✷✻✱ ✳✺✮ ✐♥tr♦ ❂ ♥✉♠♣②✳❝♦♥❝❛t❡♥❛t❡✭✭❊✶✱ ●✱ ❇✱ ❊✷✱ ❇✱ ●✮✮ ✳✳✳ s♦♥❣ ❂ ♥✉♠♣②✳❝♦♥❝❛t❡♥❛t❡✭✭✐♥tr♦✱ ✐♥tr♦✱ ✳✳✳✮✮ s❝✐t♦♦❧s✳s♦✉♥❞✳♣❧❛②✭s♦♥❣✮ s❝✐t♦♦❧s✳s♦✉♥❞✳✇r✐t❡✭s♦♥❣✱ ✬t♠♣✳✇❛✈✬✮

❙✉♠♠❛r② ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s

❙❡q✉❡♥❝❡✿ x✵✱ x✶✱ x✷✱ . . .✱ xn✱ . . .✱ xN ❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ xn✱ xn−✶ ❛♥❞ ♠❛②❜❡ xn−✷ ✭♦r ♠♦r❡ t❡r♠s ✐♥ t❤❡ ✧♣❛st✧✮ ✰ ❦♥♦✇♥ st❛rt ✈❛❧✉❡ x✵ ✭❛♥❞ ♠♦r❡ ✈❛❧✉❡s x✶✱ ✳✳✳ ✐❢ ♠♦r❡ ❧❡✈❡❧s ❡♥t❡r t❤❡ ❡q✉❛t✐♦♥✮ ❙♦❧✉t✐♦♥ ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❜② s✐♠✉❧❛t✐♦♥✿

✐♥❞❡①❴s❡t ❂ ❁❛rr❛② ♦❢ ♥✲✈❛❧✉❡s✿ ✵✱ ✶✱ ✳✳✳✱ ◆❃ ① ❂ ③❡r♦s✭◆✰✶✮ ①❬✵❪ ❂ ①✵ ❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❬♥❪ ❂ ❁❢♦r♠✉❧❛ ✐♥✈♦❧✈✐♥❣ ①❬♥✲✶❪❃

❈❛♥ ❤❛✈❡ ✭s✐♠♣❧❡✮ s②st❡♠s ♦❢ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✿

❢♦r ♥ ✐♥ ✐♥❞❡①❴s❡t❬✶✿❪✿ ①❬♥❪ ❂ ❁❢♦r♠✉❧❛ ✐♥✈♦❧✈✐♥❣ ①❬♥✲✶❪❃ ②❬♥❪ ❂ ❁❢♦r♠✉❧❛ ✐♥✈♦❧✈✐♥❣ ②❬♥✲✶❪ ❛♥❞ ①❬♥❪❃

❚❛②❧♦r s❡r✐❡s ❛♥❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s s✉❝❤ ❛s ◆❡✇t♦♥✬s ♠❡t❤♦❞ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ♦❢t❡♥ r❡s✉❧t✐♥❣ ✐♥ ❛ ❣♦♦❞ ✇❛② ♦❢ ♣r♦❣r❛♠♠✐♥❣ t❤❡ ❢♦r♠✉❧❛s

❙✉♠♠❛r✐③✐♥❣ ❡①❛♠♣❧❡✿ ♠✉s✐❝ ♦❢ s❡q✉❡♥❝❡s

• ✐✈❡♥ ❛ x✵✱ x✶✱ x✷✱ . . .✱ xn✱ . . .✱ xN

❈❛♥ ✇❡ ❧✐st❡♥ t♦ t❤✐s s❡q✉❡♥❝❡ ❛s ✧♠✉s✐❝✧❄ ❨❡s✱ ✇❡ ❥✉st tr❛♥s❢♦r♠ t❤❡ xn ✈❛❧✉❡s t♦ s✉✐t❛❜❧❡ ❢r❡q✉❡♥❝✐❡s ❛♥❞ ✉s❡ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ s❝✐t♦♦❧s✳s♦✉♥❞ t♦ ❣❡♥❡r❛t❡ t♦♥❡s ❲❡ ✇✐❧❧ st✉❞② t✇♦ s❡q✉❡♥❝❡s✿ xn = e−✹n/N s✐♥(✽πn/N) ❛♥❞ xn = xn−✶ + qxn−✶ (✶ − xn−✶) , x = x✵ ❚❤❡ ✜rst ❤❛s ✈❛❧✉❡s ✐♥ [−✶, ✶]✱ t❤❡ ♦t❤❡r ❢r♦♠ x✵ = ✵.✵✶ ✉♣ t♦ ❛r♦✉♥❞ ✶ ❚r❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ✧✉♥✐t✧ xn t♦ ❢r❡q✉❡♥❝✐❡s✿ yn = ✹✹✵ + ✷✵✵xn ✭✜rst s❡q✉❡♥❝❡ t❤❡♥ ❣✐✈❡s t♦♥❡s ❜❡t✇❡❡♥ ✷✹✵ ❍③ ❛♥❞ ✻✹✵ ❍③✮

▼♦❞✉❧❡ ✜❧❡✿ s♦✉♥❞❡q✳♣②

❚❤r❡❡ ❢✉♥❝t✐♦♥s✿ t✇♦ ❢♦r ❣❡♥❡r❛t✐♥❣ s❡q✉❡♥❝❡s✱ ♦♥❡ ❢♦r t❤❡ s♦✉♥❞ ▲♦♦❦ ❛t ❢✐❧❡s✴s♦✉♥❞❡q✳♣② ❢♦r ❝♦♠♣❧❡t❡ ❝♦❞❡ ❚r② ✐t ♦✉t ✐♥ t❤❡s❡ ❡①❛♠♣❧❡s✿

❚❡r♠✐♥❛❧❃ ♣②t❤♦♥ s♦✉♥❞s❡q✳♣② ♦s❝✐❧❧❛t✐♦♥s ✹✵ ❚❡r♠✐♥❛❧❃ ♣②t❤♦♥ s♦✉♥❞s❡q✳♣② ❧♦❣✐st✐❝ ✶✵✵

❚r② t♦ ❝❤❛♥❣❡ t❤❡ ❢r❡q✉❡♥❝② r❛♥❣❡ ❢r♦♠ ✷✵✵ t♦ ✹✵✵✳