Open O Número de Euler

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❖ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r

♣♦r

❘❛♠♦♥ ❋♦r♠✐❣❛ ❋✐❣✉❡✐r❛

s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❊❞✉❛r❞♦ ●♦♥ç❛❧✈❡s ❞♦s ❙❛♥t♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛✲ t❡♠át✐❝❛✳

❏❛♥❡✐r♦✴✷✵✶✼ ❏♦ã♦ P❡ss♦❛ ✲ P❇

† _{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙ ✲ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

F475n Figueira, Ramon Formiga.

O Número de Euler / Ramon Formiga Figueira.- João Pessoa, 2017.

79f. : il.

Orientador: Eduardo Gonçalves dos Santos Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Número de Euler. 3. Logaritmo natural. 4. Fatorial. 5. Fórmula de Stirling.

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛s ♠❛r❛✈✐❧❤❛s r❡❛❧✐③❛❞❛s ♥❛ ♠✐♥❤❛ ✈✐❞❛✳ ❊st❡ tr❛❜❛❧❤♦ é✱ ❝♦♠ ❝❡rt❡③❛✱ ♠❛✐s ✉♠❛ ♣r♦✈❛ ❞❡ q✉❡ ❊❧❡ ❡stá ♥♦ ❝♦♠❛♥❞♦✳ ❖❜r✐❣❛❞♦✱ ❙❡♥❤♦r✱ ♣♦r s❡ ❢❛③❡r ♣r❡s❡♥t❡ ❞✉r❛♥t❡ t♦❞❛ ❛ tr❛❥❡tór✐❛ q✉❡ ♠❡ tr♦✉①❡ ❛té ❛q✉✐✳

❆♦s ♠❡✉s ♣❛✐s✱ ❊❞✉❛r❞♦ ❡ ▼❛r✐❛ ●♦r❡tt❡✱ ❡ ❛ ♠✐♥❤❛ ✐r♠ã✱ ❘❡❜❡❝❛✱ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✱ ✐♥❝❡♥t✐✈❛♥❞♦✲♠❡ ❡ ❛❥✉❞❛♥❞♦✲♠❡ ❛ r❡❛❧✐③❛r ♠❡✉s s♦♥❤♦s✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ▼❛②❛r❛✱ ♣♦r s✉♣♦rt❛r ❝♦♠ ♣❛❝✐ê♥❝✐❛ ❡ s❡r❡♥✐❞❛❞❡ ❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❢r❡♥t❛❞❛s ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡ s❡r ♣❛r❛ ♠✐♠ ✉♠ ♣♦rt♦ s❡❣✉r♦✳

❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ❛♠✐❣♦s ❘ô♠✉❧♦✱ ❉❛✈✐❞✱ ❉✐❡❣♦✱ ▼❛✐❧✲ s♦♥✱ ❏♦sé ❈❛r❧♦s✱ ▼❛♥♦❡❧ ❡ ❊r✐❡❧s♦♥✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❞✉r❛♥t❡ ♥♦ss❛ ár❞✉❛ ❝❛♠✐♥❤❛❞❛ ♥♦ P❘❖❋▼❆❚✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❝✉rs♦✱ ❇r✉♥♦✱ ❈❛r❧♦s ❇♦❝❦❡r✱ ❊❧✐s❛♥❞r❛✱ ▼✐r✐❛♠✱ ❋❧❛♥❦✱ ◆❛✲ ♣♦❧❡♦♥✱ ▲✐③❛♥❞r♦ ❡ ▲❡♥✐♠❛r✱ ♣♦r t♦❞♦s ♦s ❡♥s✐♥❛♠❡♥t♦s✳

❆♦ ♣r♦❢❡ss♦r ❊❞✉❛r❞♦✱ ♣♦r t❡r ❛❝❡✐t❛❞♦ ♦ ❞❡s❛✜♦ ❞❡ s❡r ♠❡✉ ♦r✐❡♥t❛❞♦r ❡ t❡r ♠❡ ❛✉①✐❧✐❛❞♦✱ s❡♠♣r❡ ❝♦♠ ♣❛❝✐ê♥❝✐❛ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ♥❡ss❛ ❝♦♥q✉✐st❛✳

❊♥✜♠✱ ❛ t♦❞♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❉❡❞✐❝❛tór✐❛

❆♦s ♠❡✉s ♣❛✐s✱ ❊❞✉❛r❞♦ ❡ ▼❛r✐❛ ●♦r❡tt❡✳

❘❡s✉♠♦

❖ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✱ ❞❡♥♦t❛❞♦ ♣♦r e ❡ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s

◆❛t✉r❛✐s✱ ❛♣❡s❛r ❞❡ s❡r ✉♠❛ ❞❛s ❝♦♥st❛♥t❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ▼❛t❡♠át✐❝❛✱ t❛♥t♦ ♣❡❧❛ ✈❛r✐❡❞❛❞❡ ❞❡ s✉❛s ✐♠♣❧✐❝❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❛♥t♦ ♣❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✱ ♣❡r♠❛♥❡❝❡ ❞❡s❝♦♥❤❡❝✐❞♦ ♣♦r ♠✉✐t♦s✳ ➱ ❝♦♠✉♠ ❡♥❝♦♥tr❛r♠♦s ❡st✉❞❛♥t❡s ❞❡ ❊♥❣❡♥❤❛r✐❛✱ ♦✉ ❛té ♠❡s♠♦ ❞❛s ❈✐ê♥❝✐❛s ❊①❛t❛s✱ q✉❡ só t♦♠❛r❛♠ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ❞❛ ❡①✐stê♥❝✐❛ ❞♦ e ❛♣ós ✉♠ ❝✉rs♦ ❞❡ ❈á❧❝✉❧♦✳ ❚❛♠❜é♠ ♥ã♦ é ❞✐❢í❝✐❧ ♥♦s

❞❡♣❛r❛r♠♦s ❝♦♠ ❛❧✉♥♦s q✉❡✱ ♠❡s♠♦ ❛♣ós t❛❧ ❝♦♥t❛t♦✱ ♣❛r❡❝❡♠ ♥✉♥❝❛ t❡r❡♠ ♣❡r❝❡✲ ❜✐❞♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❡ ♥ú♠❡r♦✳ ❖ e é ✉♠❛ ❝♦♥st❛♥t❡ ✈❡rsát✐❧✳ ❆♣❡s❛r ❞❡✱ ❡♠

❣❡r❛❧✱ ❛♣❛r❡❝❡r r❡❧❛❝✐♦♥❛❞♦ ❛ r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❡❧❡ s❡ ❢❛③ ♣r❡s❡♥t❡ ❡♠ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❞❡ ❞✐❢❡r❡♥t❡s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✳ P♦❞❡✲ ♠♦s ❡♥❝♦♥trá✲❧♦✱ ❛❧é♠ ❞❛ ❆♥á❧✐s❡ ❡ ❚❡♦r✐❛ ❞❡ ❋✉♥çõ❡s✱ ♥❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ♥❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛✱ ♥❛ Pr♦❜❛❜✐❧✐❞❛❞❡✱ ♥❛ ❚r✐❣♦♥♦♠❡tr✐❛✱ ♥❛ ●❡♦♠❡tr✐❛✱ ♥❛ ❊st❛tíst✐❝❛✱ ♥❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛♠♦s ✉♠❛ ❜r❡✈❡ ❛♥á❧✐s❡ ❤✐stór✐❝❛ s♦❜r❡ ♦ ❞❡s❝♦❜r✐♠❡♥t♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✱ ❡①✐❜✐♠♦s s✉❛ ❞❡✜♥✐çã♦✱ ❛❧é♠ ❞❡ ❢♦r♠❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❝❛r❛❝t❡r✐③á✲❧♦ ❛tr❛✈és ❞❡ s♦♠❛s ❡ ♣r♦❞✉t♦s ✐♥✜♥✐t♦s✱ ❡ ❛❜♦r✲ ❞❛♠♦s ❞♦✐s ✐♥t❡r❡ss❛♥t❡s ♣r♦❜❧❡♠❛s ♥♦s q✉❛✐s ❡❧❡ s❡ ❢❛③ ♣r❡s❡♥t❡✿ ♦ ❞❛ ❝♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❛rt✐çõ❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ✜♥✐t♦ ❡ ♦ ❞❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ ♥♦ q✉❛❧ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ ❋ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✱ ▲♦❣❛r✐t♠♦ ◆❛t✉r❛❧✱ ❋❛t♦r✐❛❧✱ ❋ór♠✉❧❛ ❞❡ ❙t✐r✲ ❧✐♥❣✳

❆❜str❛❝t

❚❤❡ ❊✉❧❡r✬s ◆✉♠❜❡r✱ ❞❡♥♦t❡❞ ❜②e❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❜❛s❡ ♦❢ t❤❡ ◆❛t✉r❛❧

▲♦❣❛r✐t❤♠s✱ ❞❡s♣✐t❡ ❜❡✐♥❣ ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥st❛♥ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❜♦t❤ ❜② t❤❡ ✈❛r✐❡t② ♦❢ ✐ts ♠❛t❤❡♠❛t✐❝❛❧ ✐♠♣❧✐❝❛t✐♦♥s ❛♥❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ r❡♠❛✐♥s ✉♥❦♥♦✇♥ t♦ ♠❛♥② ♣❡♦♣❧❡✳ ■t ✐s ❝♦♠♠♦♥ t♦ ✜♥❞ ❊♥❣✐♥❡❡r✐♥❣ ♦r ❡✈❡♥ ❊①❛❝t ❙❝✐❡♥❝❡s st✉❞❡♥ts ✇❤♦ ♦♥❧② ❜❡❝❛♠❡ ❛✇❛r❡ ♦❢ t❤❡ ❡①✐st❡♥❝❡ ♦❢ e ❛❢t❡r

t❛❦✐♥❣ ❛ ❈❛❧❝✉❧✉s ❈♦✉rs❡✳ ■t ✐s ❛❧s♦ ♥♦t ❞✐✣❝✉❧t t♦ ✜♥❞ st✉❞❡♥ts ✇❤♦✱ ❡✈❡♥ ❛❢t❡r s✉❝❤ ❝♦♥t❛❝t✱ s❡❡♠ t♦ ♥❡✈❡r r❡❛❧✐③❡ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ♥✉♠❜❡r✳ ❚❤❡ e ✐s ❛ ✈❡rs❛t✐❧❡

❝♦♥st❛♥t✳ ❆❧t❤♦✉❣❤✱ ✐♥ ❣❡♥❡r❛❧✱ ✐t ❛♣♣❡❛rs r❡❧❛t❡❞ t♦ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s✱ ✐t ✐s ♣r❡s❡♥t ✐♥ s❡✈❡r❛❧ ♣r♦❜❧❡♠s ♦❢ ❞✐✛❡r❡♥t ▼❛t❤❡♠❛t✐❝s ❛r❡❛s✳ ❲❡ ❝❛♥ ✜♥❞ ✐t✱ ❜❡s✐❞❡s ❆♥❛❧②s✐s ❛♥❞ ❋✉♥❝t✐♦♥ ❚❤❡♦r②✱ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡✲ ♠❛t✐❝s✱ ❈♦♠❜✐♥❛t♦r✐❛❧ ❆♥❛❧②s✐s✱ Pr♦❜❛❜✐❧✐t②✱ ❚r✐❣♦♥♦♠❡tr②✱ ●❡♦♠❡tr②✱ ❙t❛t✐st✐❝s✱ ◆✉♠❜❡r ❚❤❡♦r②✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♠❛❦❡ ❛ ❜r✐❡❢ ❤✐st♦r✐❝❛❧ ❛♥❛❧②s✐s ❛❜♦✉t t❤❡ ❞✐s❝♦✲ ✈❡r② ♦❢ t❤❡ ❊✉❧❡r✬s ◆✉♠❜❡r✱ ✇❡ ♣r❡s❡♥t ✐ts ❞❡✜♥✐t✐♦♥✱ ❛s ✇❡❧❧ ❛s ❛❧t❡r♥❛t✐✈❡ ✇❛②s ♦❢ ❝❤❛r❛❝t❡r✐③✐♥❣ ✐t t❤r♦✉❣❤ ✐♥✜♥✐t❡ s✉♠s ❛♥❞ ♣r♦❞✉❝ts✳ ❲❡ ❛❧s♦ ❛❞❞r❡ss t✇♦ ✐♥t❡r❡s✲ t✐♥❣ ♣r♦❜❧❡♠s ✐♥ ✇❤✐❝❤ ✐t ✐s ♣r❡s❡♥t✿ t❤❡ ❝♦✉♥t✐♥❣ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐t✐♦♥s ♦❢ ❛ ✜♥✐t❡ ♥♦♥✲❡♠♣t② s❡t ❛♥❞ ♦❜t❛✐♥✐♥❣ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ ❢❛❝t♦r✐❛❧ ♦❢ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r✱ ✐♥ ✇❤✐❝❤ ✇❡ ✜♥❞ t❤❡ ❙t✐r❧✐♥❣✬s ❆♣♣r♦①✐♠❛t✐♦♥✳

❑❡②✇♦r❞s✿ ❊✉❧❡r✬s ◆✉♠❜❡r✱ ◆❛t✉r❛❧ ▲♦❣❛r✐t❤♠✱ ❋❛❝t♦r✐❛❧✱ ❙t✐r❧✐♥❣✬s ❆♣♣r♦①✐♠❛✲ t✐♦♥✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✈✐✐✐

✶ ❈♦♥❤❡❝❡♥❞♦ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ✶

✶✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❖ ♥ú♠❡r♦ ❡ ❝♦♠♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳✶ ■rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ ❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳✶ ❯♠❛ ❝♦♥s✐❞❡r❛çã♦ s♦❜r❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ ❡ ✳ ✳ ✳ ✳ ✶✹

✷ ❖ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ❡ ❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ✶✽

✷✳✶ ❖ ♣r♦❜❧❡♠❛ ❞❛s ♣❛rt✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸ ❯♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ✷✸ ✸✳✶ ❆ ❋ór♠✉❧❛ ❞❡ ❲❛❧❧✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❆ ❋ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ❘❡♣r❡s❡♥t❛♥❞♦ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ♣♦r ✉♠ ♣r♦❞✉t♦ ✐♥✜♥✐t♦ ✸✽ ✹✳✶ ❯♠ ♣r♦❞✉t♦ ✐♥✜♥✐t♦ ♣❛r❛ ❡♠ _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽} ✹✳✷ ❘❡✈✐s✐t❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸ ❆ ❋ór♠✉❧❛ ❞❡ P✐♣♣❡♥❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✹ ❆ ❋ór♠✉❧❛ ❞❡ ❈❛t❛❧❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✺✷

❆♣ê♥❞✐❝❡s ✺✸

❆♣ê♥❞✐❝❡ ❆ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s ✺✹

❆✳✶ ❙❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❆✳✷ ▲✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❆✳✸ Pr♦♣r✐❡❞❛❞❡s ❛r✐t♠ét✐❝❛s ❞♦s ❧✐♠✐t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ❆✳✹ ❙ér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

❆✳✺ ❖♣❡r❛çõ❡s ❝♦♠ sér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

❆♣ê♥❞✐❝❡ ❇ ❚❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❧♦❣❛rít♠✐❝❛s ✻✸

❆♣ê♥❞✐❝❡ ❈ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✻✺

❈✳✶ ❖ Pr✐♠❡✐r♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❈✳✷ ❖ ❙❡❣✉♥❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✼

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❚r❡❝❤♦ ❞❛ ♦❜r❛ ■♥tr♦❞✉❝t✐♦ ✐♥ ❛♥❛❧②s✐♥ ✐♥✜♥✐t♦r✉♠✱ ❞❡ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳ ✺ ✶✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡H₁1+x ♣❛r❛ ❡st✐♠❛t✐✈❛ ❞❡ln(1 +x)✳ ✶✸

✶✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡ Heh

1 ♣❛r❛ h >0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✶✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡ Heh

1 ♣❛r❛ h <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸✳✶ ●rá✜❝♦ ❞❡y= (1₋x2₎n _{♣❛r❛ n= 0,1/2,1,3/2,2,5/2,3,7/2,4. ✳ ✳ ✳ ✳ ✷✽}

✸✳✷ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ r❡s✉❧t❛❞♦ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❚r❡❝❤♦ ❞❛ ❝✉r✈❛y=f(x) ❧✐♠✐t❛❞♦ ♣❡❧❛s r❡t❛s x=k ❡ x=k+ 1✳ ✳ ✳ ✳ ✸✷

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶✳✶ ❈♦♠♣♦rt❛♠❡♥t♦ ❞♦ ♠♦♥t❛♥t❡ ❝♦♠ ♦ ❛✉♠❡♥t♦ ❞♦ ✈❛❧♦r ❞❡ n✳ ✳ ✳ ✳ ✳ ✳ ✷

✸✳✶ ❆♣r♦①✐♠❛♥❞♦ n! ♣❡❧❛ ❋ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

■♥tr♦❞✉çã♦

❊♠ s❛❧❛ ❞❡ ❛✉❧❛✱ é ❝♦♠✉♠✱ q✉❛♥❞♦ ❡st❛♠♦s ❛♣r❡s❡♥t❛♥❞♦ ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐✲ ❝♦s✱ q✉❡st✐♦♥❛r♠♦s ♦s ❛❧✉♥♦s s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❡ ♣❡❞✐r♠♦s ♣❛r❛ q✉❡ ❡❧❡s ♥♦s ❞❡❡♠ ❡①❡♠♣❧♦s ❞❡ t❛✐s ♥ú♠❡r♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ s✉❛ ♣ró♣r✐❛ ❡①♣❡✲ r✐ê♥❝✐❛ ♠❛t❡♠át✐❝❛✳ ◆ã♦ s✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❞❛❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ s❡✉ ❞✐â♠❡tr♦✱ ❛ q✉❛❧ ❞❡♥♦♠✐♥❛♠♦s ❞❡ π✱ s❡ ❢❛③ ♣r❡s❡♥t❡ ❡♥tr❡ ♦s ❡①❡♠♣❧♦s✱ ♥♦s q✉❛✐s t❛♠❜é♠

s❡ ✐♥❝❧✉❡♠ ♦s ♥ú♠❡r♦s √2 ❡ √3✳ ❊♠ ❛❧❣✉♠❛s t✉r♠❛s✱ ❝❤❡❣❛♠ ❛ ❝✐t❛r ♦ ◆ú♠❡r♦

❞❡ ❖✉r♦✱ φ✱ q✉❡✱ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❡st✉❞✐♦s♦s ❛ ♠❛✐s ❛❣r❛❞á✈❡❧ ♣r♦♣♦rçã♦ ❡♥tr❡ ❞♦✐s

s❡❣♠❡♥t♦s ♦✉ ♠❡❞✐❞❛s✱ é ✉t✐❧✐③❛❞♦ ♥❛ ❛rt❡ ❡ ❛rq✉✐t❡t✉r❛ ❞❡s❞❡ ❛ ❆♥t✐❣✉✐❞❛❞❡✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ▼❛t❡♠át✐❝❛ ❞✐✜❝✐❧♠❡♥t❡ s❡ ♠♦str❛ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦s ❡st✉❞❛♥t❡s✿ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ❛♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ▼❛✲ t❡♠át✐❝❛✱ s❡❥❛♠ ❡❧❡s ♣r♦❢❡ss♦r❡s✱ ❛❧✉♥♦s ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❝✉r✐♦s♦s ❛♣❛✐①♦♥❛❞♦s ♣♦r t❛❧ ❝✐ê♥❝✐❛✱ ❡ss❡ ♥ú♠❡r♦ tã♦ r❡❧❡✈❛♥t❡ q✉❡ ♣❡r♠❛♥❡❝❡ ❞❡s❝♦♥❤❡❝✐❞♦ ♣♦r ♠✉✐t♦s✱ ❝♦♥✲ t❛♥❞♦ ✉♠ ♣♦✉❝♦ ❞❛ s✉❛ ❤✐stór✐❛ ❡ ❡①✐❜✐♥❞♦ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❞❡ ❞✐❢❡r❡♥t❡s r❛♠♦s ♥♦s q✉❛✐s ❡❧❡ s❡ ❢❛③ ♣r❡s❡♥t❡✱ ♣♦r ✈❡③❡s ❛té ✐♥❡s♣❡r❛❞❛♠❡♥t❡✳ P❛r❛ ✐ss♦✱ ❡str✉t✉r❛♠♦s ♦ ❝♦♥t❡ú❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✱ ♥♦s q✉❛✐s✱ ❛❧é♠ ❞❡ ❞❡✜♥✐r♠♦s ♦ e✱ ❝♦♠♦ ♦ ◆ú♠❡r♦

❞❡ ❊✉❧❡r t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦✱ ❜✉s❝❛♠♦s r❡✈❡❧❛r s✉❛ ♣r❡s❡♥ç❛ ❛♥❛❧✐s❛♥❞♦ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ♥❛ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ tr❛t❛♠♦s ❞❡ ❛♣r❡s❡♥t❛r ❛♦ ❧❡✐t♦r✱ ❞❡ ♠❛♥❡✐r❛ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛✱ ♦ e✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❜✉s❝❛♠♦s s✐t✉❛r ♦ s✉r❣✐♠❡♥t♦ ❞❡ss❡ ♥ú♠❡r♦ ♥❛ ❤✐stór✐❛ ❞❛

▼❛t❡♠át✐❝❛✱ r❡❛❧✐③❛♥❞♦ ✉♠❛ sí♥t❡s❡ ❝♦♥t❡♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♥st✐t✉❡♠ ❡ss❛ ♣❛rt❡ ❞❛ ❤✐stór✐❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛ s✉❛ ❞❡✜♥✐çã♦ ❝♦♠♦ ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é ❞❛❞♦ ♣♦r xn = 1 + _n1n ❡ ♣r♦✈❛♠♦s q✉❡ ❡❧❡ ♣♦❞❡ s❡r

r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ sér✐❡✶ P 1

n! ♦✉✱ ❛✐♥❞❛✱ ❝❛r❛❝t❡r✐③❛❞♦ ❝♦♠♦ ❛ ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s

◆❛t✉r❛✐s✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ♣♦r ♠❡✐♦ ❞❛ sér✐❡ ❝✐t❛❞❛ é ❛ ❝❤❛✈❡ ♣r✐♥❝✐♣❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡ s✉❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❛♣r❡s❡♥t❛❞❛ ♥❡st❡ ❝❛♣ít✉❧♦✳

◆♦s ❝❛♣ít✉❧♦s q✉❡ s❡ s❡❣✉❡♠✱ ✐♥✐❝✐❛✲s❡ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♥♦s q✉❛✐s ♦ ♥ú♠❡r♦es✉r❣❡ ❞❡ ♠♦❞♦ ✐♥✉s✐t❛❞♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ♠❛♥❡✐r❛

✶_{❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛✱ ❞✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ✉t✐❧✐③❛r❡♠♦s✱ ♣♦r ✈❡③❡s✱ ❛}

♥♦t❛çã♦P 1

n! ♣❛r❛ ♥♦s r❡❢❡r✐r♠♦s à sér✐❡

∞

P

n=0

1

n!✳

❞❡ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ♣❛rt✐çõ❡s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ✜♥✐t♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❡q✉❛çã♦ ♥❛ q✉❛❧ ❛♣❛r❡❝❡ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳ ❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ♣♦r s✉❛ ✈❡③✱ é ✐♥t❡✐r❛♠❡♥t❡ ❞❡❞✐❝❛❞♦ à ❢❛♠♦s❛ ❋ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣✱ ✉♠❛ ❡①♣r❡ssã♦ ♠❛t❡♠át✐❝❛ q✉❡ ♥♦s ❞á ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡♠ ❢✉♥çã♦ t❛♥t♦ ❞❡ e q✉❛♥t♦ ❞❡ π✱ ♦ q✉❡ é ✉♠ r❡s✉❧t❛❞♦ s✉r♣r❡❡♥❞❡♥t❡✳

◆♦ q✉❛rt♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s três ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❞❡ r❡♣r❡s❡♥t❛r ♦ ♥ú♠❡r♦ e ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✐♥✜♥✐t♦✳ ❉❡♥tr❡ t❛✐s r❡♣r❡s❡♥t❛çõ❡s ❡♥❝♦♥tr❛♠✲s❡

❞✉❛s ✐♥t❡r❡ss❛♥t❡s ❡①♣r❡ssõ❡s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ P✐♣♣❡♥❣❡r ❡ ❋ór♠✉❧❛ ❞❡ ❈❛t❛❧❛♥✳

❈❛♣ít✉❧♦ ✶

❈♦♥❤❡❝❡♥❞♦ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r

❊st❡ ❝❛♣ít✉❧♦ é ❞❡st✐♥❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳ ◆❡❧❡✱ ❝♦♥t❛♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ q✉❡ ❡♥✈♦❧✈❡ ♦ s✉r❣✐♠❡♥t♦ ❞❡ss❡ ♥ú♠❡r♦ ❡ ❛♣r❡s❡♥t❛♠♦s s✉❛ ❞❡✜♥✐çã♦ ❡ ❛❧❣✉♠❛s ♦✉tr❛s ❢♦r♠❛s ❞❡ ❝❛r❛❝t❡r✐③á✲❧♦✳

✶✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ ♥ú♠❡r♦ π✱ ❞♦ q✉❛❧ ❥á s❡ t✐♥❤❛ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡s❞❡ ❛ ❆♥t✐✲

❣✉✐❞❛❞❡✱ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✱ ❞❡♥♦t❛❞♦ ♣♦r e ❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛ 2,71828✱

só ✈❡✐♦ ❛ s❡r ❞❡s❝♦❜❡rt♦ ♥❛ ■❞❛❞❡ ▼♦❞❡r♥❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ▼❛♦r ✭✶✾✾✹✱ ♣✳✶✻✮✱ ♦ ♣r✐♠❡✐r♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❡①♣❧í❝✐t♦ ❞♦ ♣❛♣❡❧ ❞♦ ♥ú♠❡r♦ e ♥❛ ▼❛t❡♠át✐❝❛ ♣❛r❡❝❡ t❡r

s✐❞♦ ❢❡✐t♦ ❡♠ ✶✻✶✽✱ ♥❛ s❡❣✉♥❞❛ ❡❞✐çã♦ ❞❛ tr❛❞✉çã♦ ❞❡ ❊❞✇❛r❞ ❲r✐❣❤t ♣❛r❛ ❛ ♦❜r❛ ▼✐r✐✜❝✐ ❧♦❣❛r✐t❤♠♦r✉♠ ❝❛♥♦♥✐s ❞❡s❝r✐♣t✐♦✶ _{❞❡ ❏♦❤♥ ◆❛♣✐❡r✱ ♦ ✐♥✈❡♥t♦r✱ ♦✉ ♠❡❧❤♦r✱} ❞❡s❝♦❜r✐❞♦r ❞♦s ❧♦❣❛r✐t♠♦s✳

❖ ❝♦♥t❡①t♦ ❞❡ ♥❛s❝✐♠❡♥t♦ ❞♦ ❝❛♣✐t❛❧✐s♠♦ ❡ ❝♦♥s❡q✉❡♥t❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ❝♦♠ér❝✐♦ ✐♥t❡r♥❛❝✐♦♥❛❧ ♥❛ ■❞❛❞❡ ▼♦❞❡r♥❛✱ ♠✉✐t♦ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ❢♦✐ ♦ ❛❣❡♥t❡ ♠♦t✐✈❛❞♦r ♣❛r❛ ❛ ❞❡s❝♦❜❡rt❛ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✱ ❛♣❡s❛r ❞❡✱ ♥❛ ♠❡s♠❛ é♣♦❝❛✱ ♦✉tr❛s q✉❡stõ❡s✱ ❝♦♠♦ ❛ q✉❛❞r❛t✉r❛ ❞❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛✱ ❝♦♥❞✉③✐r❡♠ ❛♦ ♠❡s♠♦ ♥ú♠❡r♦✳ ▼❛s✱ ❝♦♠♦ ❡ss❡ ❝r❡s❝✐♠❡♥t♦ ❝♦♠❡r❝✐❛❧ ♠♦t✐✈♦✉ ♦ s✉r❣✐♠❡♥t♦ ❞♦ e❄

❙❡❣✉♥❞♦ ▼❛♦r ✭✶✾✾✹✱ ♣✳✷✻✮✱ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ♣♦❞❡r✐❛ ❡st❛r ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞♦ ❛ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s✳ ❙❡ ✉♠ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧ ❞❡ ❘✩ ✶✱✵✵ ❢♦r ✐♥✈❡st✐❞♦ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s ❛♥✉❛❧ ❞❡ ✶✵✵✪ ❝❛♣✐t❛❧✐③❛❞♦s ❛♥✉❛❧♠❡♥t❡✱ ❛♦ ✜♠ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ♦ ♠♦♥t❛♥t❡ ♦❜t✐❞♦ s❡rá ❞❛❞♦ ♣♦rM = (1 + 1)1 = 2✳ ❈❛s♦ ❛ ❝❛♣✐t❛❧✐③❛çã♦ ❢♦ss❡ r❡❛❧✐③❛❞❛ s❡♠❡str❛❧♠❡♥t❡✱ ❡ss❡ ✈❛❧♦r ♣❛ss❛r✐❛ ❛ s❡rM = 1 + 1₂2 = 2,25✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s❡ ❛ ❝❛♣✐t❛❧✐③❛çã♦ ♦❝♦rr❡ss❡ ❛ ❝❛❞❛ tr✐♠❡str❡✱

t❡rí❛♠♦s M = 1 +1₄4 ≈ 2,44✳ ❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ r❡❛❧✐③❛♥❞♦ ❛ ❝❛♣✐t❛❧✐③❛çã♦ n

✶_{❊♠ ✉♠ ❞♦s ❛♣ê♥❞✐❝❡s ❞❡st❛ ♦❜r❛✱ ❛♣❛r❡❝❡ ♦ ❡q✉✐✈❛❧❡♥t❡ ❞❛ ❞❡❝❧❛r❛çã♦ ❞❡ q✉❡ loge10 =}

2,302585✭▼❆❖❘✱ ✶✾✾✹✱ ♣✳✶✻✮✳

✶✳✶✳ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

✈❡③❡s ❡♠ ✉♠ ❛♥♦✱ ♦❜t❡rí❛♠♦s M = 1 + 1

n

n

✳ ➱ ❡st❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❛♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳

▼❡s♠♦ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡✱ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦✱ só t❡♥❤❛ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦st❡r✐♦r♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳❱■■✱ ♣♦r ♠❡✐♦ ❞♦s tr❛❜❛❧❤♦s ❞❡ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✱ é ♣r♦✈á✈❡❧ q✉❡ ♥❛ é♣♦❝❛ ❞❡ ◆❛♣✐❡r✱ ✐♥í❝✐♦ ❞❡ss❡ ♠❡s♠♦ sé❝✉❧♦✱ ❛❧❣✉é♠ ❥á t❡♥❤❛ s❡ ♣❡r❣✉♥t❛❞♦ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ M q✉❛♥❞♦ ❛✉♠❡♥t❛♠♦s ✐♥❞❡✲

✜♥✐❞❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❡ n✳ ❖ ♣r♦❝❡ss♦ ❞❡ ✈❡r✐✜❝❛çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦

q✉❡ r❡♣r❡s❡♥t❛ ♦ ♠♦♥t❛♥t❡ à ♠❡❞✐❞❛ q✉❡ n ❝r❡s❝❡ ❝♦♥❞✉③✐✉ ♦s ♠❛t❡♠át✐❝♦s ❛♦ ❡♥✲

❝♦♥tr♦ ❞♦ e✳ P♦r ♠❡✐♦ ❞❛ ❚❛❜❡❧❛ ✶✳✶ é ♣♦ssí✈❡❧ ✐♥❢❡r✐r ❡ss❡ ❝♦♠♣♦rt❛♠❡♥t♦✳ ❈♦♠♦

♦s ✈❛❧♦r❡s ❞❡ M ❛♣r❡s❡♥t❛❞♦s sã♦ ❛♣r♦①✐♠❛çõ❡s ❝♦♠✱ ♥♦ ♠á①✐♠♦✱ ✺ ❝❛s❛s ❞❡❝✐♠❛✐s✱

♥ã♦ ❝♦♥s❡❣✉✐♠♦s ❡♥①❡r❣❛r ❛ ✈❛r✐❛çã♦ q✉❡ ♦❝♦rr❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ♣❛ss❛♠♦s ❞❡ n = 1000000 ♣❛r❛ n = 10000000 ❡✱ ✐♥t✉✐t✐✈❛♠❡♥t❡✱ s♦♠♦s t❡♥t❛❞♦s ❛ ❝♦♥❝❧✉✐r

q✉❡ ❝♦♠ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ✈❛❧♦r ❞❡ n ❡ss❛ ✈❛r✐❛çã♦ t❡♥❞❡rá ❛ ❛❝♦♥t❡❝❡r ❡♠ ❝❛s❛s

❞❡❝✐♠❛✐s ❝❛❞❛ ✈❡③ ♠❛✐s ❞✐st❛♥t❡s ❞❛ ✈ír❣✉❧❛✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ é ♥❛t✉r❛❧ ♣❡♥s❛r q✉❡✱ ♠❡s♠♦ ❛✉♠❡♥t❛♥❞♦ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❡n✱ ♦ ♠♦♥t❛♥t❡ r❡s✉❧t❛rá ❡♠ ✉♠

♥ú♠❡r♦ r❡❛❧ ❜❡♠ ❞❡✜♥✐❞♦✱ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳

n M = 1 + 1

n

n

1 2

2 2,25

3 2,37037

4 2,44141

5 2,48832

10 2,59374

50 2,69159

100 2,70481

1000 2,71692

10000 2,71815

100000 2,71827

1000000 2,71828

10000000 2,71828

❚❛❜❡❧❛ ✶✳✶✿ ❈♦♠♣♦rt❛♠❡♥t♦ ❞♦ ♠♦♥t❛♥t❡ ❝♦♠ ♦ ❛✉♠❡♥t♦ ❞♦ ✈❛❧♦r ❞❡ n✳

✶✳✶✳ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

❈♦♠♦ ❛✜r♠❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ♦s tr❛❜❛❧❤♦s r❡❧❛❝✐♦♥❛❞♦s à q✉❛❞r❛t✉r❛ ❞❛ ❤✐✲ ♣ér❜♦❧❡ ❡q✉✐❧át❡r❛✱ ✐st♦ é✱ ❛♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛y = 1

x✱ t❛♠❜é♠ ❝♦♥❞✉③✐r❛♠

♦s ♠❛t❡♠át✐❝♦s ❞♦ sé❝✉❧♦ ❳❱■■ ❛ s❡ ❞❡♣❛r❛r❡♠ ❝♦♠ ♦ e✳ ❖ ❝♦♥❤❡❝✐❞♦ ♠❛t❡♠át✐❝♦

P✐❡rr❡ ❞❡ ❋❡r♠❛t✱ ❡♠ t♦r♥♦ ❞♦ ❛♥♦ ✶✻✹✵ ✭❝❡r❝❛ ❞❡ tr✐♥t❛ ❛♥♦s ❛♥t❡s ❞♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ♣♦r ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✮✱ ❞❡♠♦♥str♦✉ q✉❡ ❛ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣❡❧❛s r❡t❛s x= 0 ❡ x =a✱ ♣❡❧♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡ ♣❡❧❛ ❝✉r✈❛ ❞❡ ❡q✉❛çã♦ y = xn_✱

❝♦♠ n ₆= ₋1✱ é ❞❛❞❛ ♣♦r an+1

n+1✳ ❊ss❡ r❡s✉❧t❛❞♦ ❢♦✐ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ♣♦✐s ♣♦ss✐❜✐❧✐✲

t❛✈❛ ❛ q✉❛❞r❛t✉r❛ ♥ã♦ s♦♠❡♥t❡ ❞❡ ✉♠❛ ❝✉r✈❛✱ ♠❛s ❞❡ t♦❞❛ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s✳ ❆♣❡s❛r ❞✐ss♦✱ ❛ ❤✐♣ér❜♦❧❡ y= _x1 ♣❡r♠❛♥❡❝❡✉ ❢♦r❛ ❞❡ss❛ ❣r❛♥❞❡ ❢❛♠í❧✐❛ ❝♦♥t❡♠♣❧❛❞❛✱

❥á q✉❡ ♣❛r❛ n=₋1♦ ❞❡♥♦♠✐♥❛❞♦r n+ 1❞❛ ❡①♣r❡ssã♦ s❡ t♦r♥❛ ✐❣✉❛❧ ❛ ✵✳ ❈♦♠♦ ❞✐t♦

♣♦r ▼❛♦r ✭✶✾✾✹✱ ♣✳✻✻✮✱ ❛ ❢r✉str❛çã♦ ❞❡ ❋❡r♠❛t ♣♦r s✉❛ ❡①♣r❡ssã♦ ♥ã♦ t❡r ❝♦❜❡rt♦ ❡st❡ ❝❛s♦ tã♦ ✐♠♣♦rt❛♥t❡ ❞❡✈❡ t❡r s✐❞♦ ❣r❛♥❞❡✳

❈♦✉❜❡ ❛ ✉♠ ❝♦♥t❡♠♣♦râ♥❡♦ ❞❡ ❋❡r♠❛t✱ ●ré❣♦✐r❡ ❞❡ ❙❛✐♥t✲❱✐♥❝❡♥t✱ r❡s♦❧✈❡r✱ ♣❡❧♦ ♠❡♥♦s ❡♠ ♣❛rt❡✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ q✉❛❞r❛t✉r❛ ❞❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛✳ ❊♠ s❡✉ tr❛✲ ❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ❖♣✉s ❣❡♦♠❡tr✐❝✉♠ q✉❛❞r❛t✉r❛❡ ❝✐r❝✉❧✐ ❡t s❡❝t✐♦♥✉♠ ❝♦♥✐✱ ●ré❣♦✐r❡ ♠♦str♦✉ q✉❡ ❛ ár❡❛ s♦❜ ❛ ❤✐♣ér❜♦❧❡ ❞❡ ❡q✉❛çã♦ y = 1

x ♥✉♠ ✐♥t❡r✈❛❧♦ [a, b] é ✐❣✉❛❧ à

ár❡❛ s♦❜ ❡st❛ ♠❡s♠❛ ❝✉r✈❛ ♥✉♠ ✐♥t❡r✈❛❧♦ [c, d]✱ s❡ a b =

c

d✳ ❆ss✐♠✱ s❡ ♣❡r❝♦rr❡r♠♦s

♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ s✐t✉❛❞♦ ❛ ✉♠❛ ❞✐s✲ tâ♥❝✐❛ d ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝❡rt❛ r❡❢❡rê♥❝✐❛✱ ❞✐❣❛♠♦s ♦ ♣♦♥t♦ x = 1✱ ❛♦ ❞♦❜r❛r♠♦s

♣r♦❣r❡ss✐✈❛♠❡♥t❡ ❡ss❛ ❞✐stâ♥❝✐❛✱ ✐st♦ é✱ ❛♦ ♣❛ss❛r♠♦s ♣❡❧♦s ♣♦♥t♦s ❝✉❥❛s ❞✐stâ♥❝✐❛s ❡♠ r❡❧❛çã♦ ❛ x = 1 sã♦ 2d,4d,8d,16d, ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛ y = 1

x ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ✶ ❛té ♦s ♣♦♥t♦s ❝✐t❛❞♦s ♣❛ss❛ ❛ s❡r 2A,3A,4A,5A✱ ❡ ❛ss✐♠

♣♦r ❞✐❛♥t❡ ✭❝♦♥s✐❞❡r❛♥❞♦✱ ♦❜✈✐❛♠❡♥t❡✱ q✉❡ ❛ ár❡❛ s♦❜ ❛ ❤✐♣ér❜♦❧❡ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ✶ ❛♦ ♣♦♥t♦ ❞❡ ❞✐stâ♥❝✐❛ d é ❞❛❞❛ ♣♦r A✮✳ ❉❡st❛ ❢♦r♠❛✱ é ♣♦ssí✈❡❧ ♦❜s❡r✈❛r q✉❡

à ♠❡❞✐❞❛ q✉❡ ❛s ❞✐stâ♥❝✐❛s ❝r❡s❝❡♠ ❡♠ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✱ ❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛ ❝r❡s❝❡ ❡♠ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ❊st❡ r❡s✉❧t❛❞♦ ✐♠♣❧✐❝❛ q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ár❡❛ ❡ ❛ ❞✐stâ♥❝✐❛ é ❧♦❣❛rít♠✐❝❛✳ ❋♦✐ ❥✉st❛♠❡♥t❡ ♣❛r❛ ❡①♣r❡ss❛r ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡ss❛ r❡❧❛çã♦ q✉❡ ✉♠ ❞♦s ❛❧✉♥♦s ❞❡ ❙❛✐♥t✲❱✐♥❝❡♥t✱ ❆❧❢♦♥s♦ ❆♥t♦♥ ❞❡ ❙❛r❛s❛✱ ❢❡③ ✉s♦✱ t❛❧✈❡③ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ❤✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❞❡ ✉♠❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ✭❛té ❡♥tã♦✱ ♦s ❧♦❣❛r✐t♠♦s ❡r❛♠ ❝♦♥s✐❞❡r❛❞♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❝á❧❝✉❧♦✮ ✭▼❆❖❘✱ ✶✾✾✹✱ ♣✳✻✼✮✳

▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ r❡s✉❧t❛❞♦ ❞❡ ❙❛✐♥t✲❱✐♥❝❡♥t ❡ ❞❡♥♦t❛♥❞♦ ♣♦r A(t) ❛

ár❡❛ s♦❜ ❛ ❤✐♣ér❜♦❧❡ ❝♦♠♣r❡❡♥❞✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡x= 1❛té ✉♠ ♣♦♥t♦ ✈❛r✐á✈❡❧x=t✱

♣♦❞❡♠♦s ❡s❝r❡✈❡r A(t) = logt✱ ♦♥❞❡log ♥ã♦ r❡♣r❡s❡♥t❛ ♦ ❧♦❣❛r✐t♠♦ ❞❡ ❜❛s❡ ✶✵✱ ♠❛s

✉♠ ❧♦❣❛r✐t♠♦ ❞❡ ❜❛s❡ ❞❡s❝♦♥❤❡❝✐❞❛✳ ❖ q✉❡ ❆❧❢♦♥s♦ ❢❡③ ❢♦✐ ❥✉st❛♠❡♥t❡ ❡s❝r❡✈❡r ✉♠❛ ❡①♣r❡ssã♦ ❞❡ss❡ t✐♣♦✱ ❡✱ ❛ss✐♠ ❝♦♠♦ ✜③❡♠♦s✱ ❡❧❡ ♥ã♦ ❡①♣❧✐❝✐t♦✉ q✉❛❧ s❡r✐❛ ❛ ❜❛s❡ ❞♦ ❧♦❣❛r✐t♠♦ ✉t✐❧✐③❛❞♦✳ ❆ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦s sé❝✉❧♦s ❳❱■■ ❡ ❳❱■■■ ❡♥❝❛rr❡❣♦✉✲s❡ ❞❡ r❡✈❡❧❛r q✉❡ t❛❧ ❜❛s❡ ❝♦rr❡s♣♦♥❞✐❛ ❛♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳

❉✐✈❡rs♦s ♦✉tr♦s ❣r❛♥❞❡s ♥♦♠❡s ❞❛ ▼❛t❡♠át✐❝❛ ❞❡✐①❛r❛♠ s✉❛ ♠❛r❝❛ ♥❛ ❤✐stór✐❛✱ q✉❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❝♦♥str✉í❞❛✱ ❞♦ ♥ú♠❡r♦ e✳ ❙❡r✐❛ ✉♠ ❞❡s❧❡✐①♦ ❞❡✐①❛r ❞❡ ❝✐t❛r ♦s

✶✳✶✳ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

♠❛t❡♠át✐❝♦s ❏❛❝♦❜ ❇❡r♥♦✉❧❧✐ ✭✶✻✺✹✲✶✼✵✺✮✱ q✉❡✱ ❡♠ ✶✻✽✸✷_{✱ ❡st✉❞❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛} ❝❛♣✐t❛❧✐③❛çã♦ ❝♦♥tí♥✉❛✱ ♠♦str♦✉ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ 1 + 1

n

n

q✉❛♥❞♦ n t❡♥❞❡ ❛ ✐♥✜♥✐t♦

s❡ ❡♥❝♦♥tr❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s ✷ ❡ ✸✱ ❡ ▲❡✐❜♥✐③ ✭✶✻✹✻✲✶✼✶✻✮✱ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ♣r✐♠❡✐r❛ ❛♣❛r✐çã♦ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛ ❞♦ ♥ú♠❡r♦ e✱ ❡♠ ✶✻✾✵✸✳ ❆♣❡s❛r ❞❡ ❡①✐st✐r❡♠ ♠✉✐t♦s

♦✉tr♦s ♠❛t❡♠át✐❝♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ✐♠❡♥s❛♠❡♥t❡ ♣❛r❛ q✉❡ ♦ ♥ú♠❡r♦e ✈✐❡ss❡ ❛ s❡

t♦r♥❛r tã♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛ q✉❛♥t♦ é ❤♦❥❡✱ ♣❛r❛ ♦s ✜♥s ❞❡st❡ tr❛❜❛❧❤♦✱ é s✉✜❝✐❡♥t❡ q✉❡ ❢❛❧❡♠♦s ❞❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❛♣❡♥❛s ♠❛✐s ✉♠ ❞❡❧❡s✿ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳ ❊✉❧❡r é ✉♠❛ ✜❣✉r❛ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ❞✐s♣❡♥s❛ ❝♦♠❡♥tár✐♦s✳ ❯♠❛ ❜r❡✈❡ ♣❡s✲ q✉✐s❛ ♥❛ ✐♥t❡r♥❡t é s✉✜❝✐❡♥t❡ ♣❛r❛ r❡✈❡❧❛r✱ ❛té ♠❡s♠♦ ❛♦ ♠❛✐s ❞❡s❛t❡♥t♦ ❧❡✐t♦r✱ ♦ q✉ã♦ ✐♠♣♦rt❛♥t❡ ❡st❡ ❤♦♠❡♠ ❢♦✐ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❛ ❝✐ê♥❝✐❛ ✭s❡ ♣❡sq✉✐s❛r✲ ♠♦s r❛♣✐❞❛♠❡♥t❡ ♣♦r ❧✐st❛s ❝♦♥t❡♥❞♦ ♦s ❞❡③ ♠❛t❡♠át✐❝♦s ♠❛✐s ✐♥✢✉❡♥t❡s ❞❡ t♦❞♦s ♦s t❡♠♣♦s✱ é ♠✉✐t♦ ♣r♦✈á✈❡❧ q✉❡ ❡❧❡ ❡st❡❥❛ ♥♦ t♦♣♦ ❡♠ t♦❞❛s ❡❧❛s✮✳ ◆❛s❝✐❞♦ ❡♠ ✶✼✵✼✱ ♥❛ ❝✐❞❛❞❡ s✉íç❛ ❞❡ ❇❛s✐❧❡✐❛✱ ❞✉r❛♥t❡ s❡✉s ✼✻ ❛♥♦s ❞❡ ✈✐❞❛✱ ❊✉❧❡r ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ t❛♥t♦ ♣✉r❛ q✉❛♥t♦ ❛♣❧✐❝❛❞❛✱ ❛❧é♠ ❞❛ ❋ís✐❝❛ ❡ ❞❛ ❆str♦♥♦♠✐❛✱ ❝❤❡❣❛♥❞♦ ❛ ♣✉❜❧✐❝❛r ♠❛✐s ❞❡ ✺✵✵ ❛rt✐❣♦s ✭❇❖❨❊❘✱ ✷✵✶✵✱ ♣✳✸✵✹✮✳ ❆❧é♠ ❞❡ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ❡♠ t❡r♠♦s ❞❡ ❝♦♥t❡ú❞♦✱ ❡❧❡ ❢♦✐ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s q✉❡ ♠❛✐s ❡①❡r❝❡r❛♠ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ❛s ♥♦t❛çõ❡s q✉❡ sã♦ ✉t✐❧✐③❛❞❛s ❤♦❞✐❡r♥❛♠❡♥t❡✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❇♦②❡r ✭✷✵✶✵✱ ♣✳✸✵✺✮✱ ❊✉❧❡r ✏❢♦✐ ♦ ❝♦♥str✉t♦r ❞❡ ♥♦✲ t❛çã♦ ♠❛✐s ❜❡♠✲s✉❝❡❞✐❞♦ ❡♠ t♦❞♦s ♦s t❡♠♣♦s✑✳ ❋♦✐ ❡❧❡ q✉❡♠ ✉t✐❧✐③♦✉ ♣r✐♠❡✐r❛♠❡♥t❡ ♦ sí♠❜♦❧♦ i ♣❛r❛ r❡♣r❡s❡♥t❛r √₋1 ❡ t♦r♥♦✉ ❧❛r❣❛♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ♦ ✉s♦ ❞❛ ❧❡tr❛ π

♣❛r❛ ❡①♣r❡ss❛r ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ♦ s❡✉ ❞✐â♠❡tr♦✱ ❛♣❡s❛r ❞❡ ♥ã♦ t❡r s✐❞♦ ♦ ♣r✐♠❡✐r♦ ❛ ✉t✐❧✐③❛r ❡ss❛ ♥♦t❛çã♦✳ ❖ ✉s♦ ❞❛ ❧❡tr❛ P _♣❛r❛

✐♥❞✐❝❛r ✉♠ s♦♠❛tór✐♦ ❡ ❞♦ sí♠❜♦❧♦ f(x)♣❛r❛ ✉♠❛ ❢✉♥çã♦ ❞❡ xt❛♠❜é♠ sã♦ ❞❡✈✐❞♦s

❛ ❊✉❧❡r✳

❆té ❛q✉✐✱ ❞✉r❛♥t❡ t♦❞♦ ♦ t❡①t♦✱ ✉t✐❧✐③❛♠♦s ❛ ❡①♣r❡ssã♦ ✏◆ú♠❡r♦ ❞❡ ❊✉❧❡r✑ ♦✉ ♦ sí♠❜♦❧♦ ✏e✑ ♣❛r❛ tr❛t❛r ❞❡ ✉♠ ♥ú♠❡r♦ q✉❡✱ ❝♦♠♦ ❞✐s❝✉t✐♠♦s✱ ❝♦rr❡s♣♦♥❞❡ ❛

lim n→∞ 1 +

1

n

n

♦✉ à ❜❛s❡ ❞♦s ❧♦❣❛r✐t♠♦s ♥❛t✉r❛✐s✳ ❖ t❡r♠♦ ✏◆ú♠❡r♦ ❞❡ ❊✉❧❡r✑ ♥ã♦ é ✉t✐❧✐③❛❞♦ ♣♦r ❛❝❛s♦✳ ❆♣❡s❛r ❞❡✱ ♥❛ ❤✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ♦ ♥ú♠❡r♦ q✉❡ é t❡♠❛ ❞❡st❛ ❞✐ss❡rt❛çã♦ t❡r s✐❞♦ ❞❡s❝♦❜❡rt♦ ♥♦ sé❝✉❧♦ ❳❱■■✱ s♦♠❡♥t❡ ♥♦ sé❝✉❧♦ ❳❱■■■✱ ❛♣ós ❊✉❧❡r t❡r ❡♠♣r❡❣❛❞♦ ♦ sí♠❜♦❧♦ e ♣❛r❛ s❡ r❡❢❡r✐r ❛ ❡❧❡✱ s✉r❣✐✉ ✉♠❛ ♥♦t❛çã♦ ♣❛❞r♦♥✐✲

③❛❞❛ ♣❛r❛ r❡♣r❡s❡♥tá✲❧♦✳ ❙❡❣✉♥❞♦ ❇♦②❡r ✭✷✵✶✵✱ ♣✳✸✵✺✮✱ ❡♠ ✉♠❛ ❡①♣♦s✐çã♦ ♠❛♥✉s❝r✐t❛ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛s s♦❜r❡ ❞✐s♣❛r♦ ❞❡ ❝❛♥❤õ❡s✱ ❡♠ ✶✼✷✼ ♦✉ ✶✼✷✽✱ ❊✉❧❡r ✉t✐✲ ❧✐③♦✉ ❛ ❧❡tr❛ e ✏♠❛✐s ❞❡ ✉♠❛ ❞ú③✐❛ ❞❡ ✈❡③❡s ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ❜❛s❡ ❞♦ s✐st❡♠❛ ❞❡

❧♦❣❛r✐t♠♦s ♥❛t✉r❛✐s✑✳ ❆❧é♠ ❞✐ss♦✱ ❡♠ ✉♠❛ ❝❛rt❛ ❛ ●♦❧❞❜❛❝❤ ❡♠ ✶✼✸✶✱ ♦ ♠❛t❡♠át✐❝♦ ✉s♦✉ ♦ e ♣❛r❛ ❡①♣r❡ss❛r ✏❛q✉❡❧❡ ♥ú♠❡r♦ ❝✉❥♦ ❧♦❣❛r✐t♠♦ ❤✐♣❡r❜ó❧✐❝♦ ❂ ✶✑ ✭❇❖❨❊❘✱

✷_{◆❡ss❛ é♣♦❝❛ ❛ ❝♦♥❡①ã♦ ❡♥tr❡ lim}

n→∞

1 + _n1n ❡ ♦s ❧♦❣❛r✐t♠♦s ♥❛t✉r❛✐s ❛✐♥❞❛ ♥ã♦ ❤❛✈✐❛ s✐❞♦ ✐❞❡♥t✐✜❝❛❞❛✳

✸_{❆♣❡s❛r ❞♦ e✱ ❥á t❡r ❛♣❛r❡❝✐❞♦✱ ❝♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡♠ tr❛❜❛❧❤♦s ❞♦ ✐♥í❝✐♦ ❞♦}

sé❝✉❧♦ ❳❱■■✱ ❢♦✐ ♥✉♠❛ ❝❛rt❛ ❞❡ ▲❡✐❜♥✐③ ❡♥❞❡r❡ç❛❞❛ ❛ ❍✉②❣❡♥s✱ q✉❡ s❡ ✉s♦✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ✉♠❛ ♥♦t❛çã♦ ♣❛r❛ ❡❧❡✱ r❡✈❡❧❛♥❞♦ q✉❡ ♥❛q✉❡❧❛ é♣♦❝❛ ❡❧❡ ❥á ❡r❛ ❝❧❛r❛♠❡♥t❡ r❡❝♦♥❤❡❝✐❞♦✳ ❊♠ s✉❛ ❝❛rt❛✱ ▲❡✐❜♥✐③ ✉t✐❧✐③♦✉ ❛ ❧❡tr❛b♣❛r❛ ❞❡♥♦tá✲❧♦✳

✶✳✷✳ ❉❡✜♥✐çã♦

✷✵✶✵✱ ♣✳✸✵✺✮✳ ❊♠ s✉❛s ♣✉❜❧✐❝❛çõ❡s✱ ❊✉❧❡r t❛♠❜é♠ ♥ã♦ ❛❜r✐❛ ♠ã♦ ❞❡ ✉t✐❧✐③❛r ❛ ❧❡tr❛e

♣❛r❛ s❡ r❡❢❡r✐r à ❜❛s❡ ❞♦s ❧♦❣❛r✐t♠♦s ♥❛t✉r❛✐s✳ ❋♦✐ ❡♠ s✉❛ ♦❜r❛ ✐♥t✐t✉❧❛❞❛ ▼❡❝❤❛♥✐❝❛✱ ❞❡ ✶✼✸✻✱ q✉❡ ♦ e ❛♣❛r❡❝❡✉ ✐♠♣r❡ss♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ✭❇❖❨❊❘✱ ✷✵✶✵✱ ♣✳✸✵✺✮✳ ◆❛

❋✐❣✉r❛ ✶✳✶✱ é ❛♣r❡s❡♥t❛❞♦ ✉♠ tr❡❝❤♦ ❞❡ ✉♠❛ ❡❞✐çã♦ ❞❡ ✶✾✷✷ ❞❡ ✉♠❛ ❞❡ s✉❛s ♦❜r❛s ♠❛✐s ❝♦♥❤❡❝✐❞❛s✱ ❛ ■♥tr♦❞✉❝t✐♦ ✐♥ ❛♥❛❧②s✐♥ ✐♥✜♥✐t♦r✉♠✹_{✱ ♣✉❜❧✐❝❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③} ❡♠ ✶✼✹✽✱ ♥♦ q✉❛❧ ❡❧❡ ✉t✐❧✐③❛ ❛ ♥♦t❛çã♦ e❡ ❛♣r❡s❡♥t❛ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❝♦♠ ✷✸ ❝❛s❛s

❞❡❝✐♠❛✐s ♣❛r❛ ❡ss❡ ♥ú♠❡r♦✱ ♦❜t✐❞❛ ♣♦r ♠❡✐♦ ❞❛ sér✐❡ P 1

n!✳

❋✐❣✉r❛ ✶✳✶✿ ❚r❡❝❤♦ ❞❛ ♦❜r❛ ■♥tr♦❞✉❝t✐♦ ✐♥ ❛♥❛❧②s✐♥ ✐♥✜♥✐t♦r✉♠✱ ❞❡ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳

✶✳✷ ❉❡✜♥✐çã♦

❆ ♣❛rt✐r ❞❡st❛ s❡çã♦✱ ❞❡✐①❛♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ❧❛❞♦ ❛ ❤✐stór✐❛ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ❡ ♣❛ss❛♠♦s ❛ tr❛t❛r ❞♦ ❝á❧❝✉❧♦ ♠❛t❡♠át✐❝♦ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦✳ ❆q✉✐✱ ♠♦str❛♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é ❞❛❞♦ ♣♦r xn = 1 +_n1n ♣♦ss✉✐ ❧✐♠✐t❡ q✉❛♥❞♦ n

t❡♥❞❡ ❛ ✐♥✜♥✐t♦✱ ❡ ❞❡✜♥✐♠♦s ♦ ♥ú♠❡r♦ e ❝♦♠♦ s❡♥❞♦ ❥✉st❛♠❡♥t❡ ❡ss❡ ❧✐♠✐t❡✳ P❛r❛

✐st♦✱ ✉t✐❧✐③❛♠♦s ♦ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ❞❛ ❆♥á❧✐s❡ ▼❛t❡♠át✐❝❛ ♦ q✉❛❧ ❣❛r❛♥t❡ q✉❡

✹_{◆❡st❛ ♦❜r❛✱ ❊✉❧❡r ❞❡♠♦♥str♦✉ q✉❡ ♦ ♥ú♠❡r♦ e✱ ❝♦♥s✐❞❡r❛❞♦ ❛ ❜❛s❡ ❞♦s ❧♦❣❛r✐t♠♦s ♥❛t✉r❛✐s✱}

t❛♠❜é♠ ❝♦rr❡s♣♦♥❞✐❛ ❛♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ 1 +n1 n

q✉❛♥❞♦nt❡♥❞❡ ❛ ✐♥✜♥✐t♦✱ ❡ ♣♦❞✐❛ s❡r ♦❜t✐❞♦

♣♦r ♠❡✐♦ ❞❛ sér✐❡1 +_1!1 +_2!1 +_3!1 +_{· · ·}✳

✶✳✷✳ ❉❡✜♥✐çã♦

t♦❞❛ s❡q✉ê♥❝✐❛ ♠♦♥ót♦♥❛ ❡ ❧✐♠✐t❛❞❛ é ❝♦♥✈❡r❣❡♥t❡✳ ❆♦ ❧❡✐t♦r ♠❡♥♦s ❢❛♠✐❧✐❛r✐③❛❞♦ t❛♥t♦ ❝♦♠ ♦ r❡s✉❧t❛❞♦ ❝✐t❛❞♦ q✉❛♥t♦ ❝♦♠ ♦s t❡r♠♦s ✏s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛✑✱ ✏❧✐♠✐t❡✑✱ ✏s❡q✉ê♥❝✐❛ ♠♦♥ót♦♥❛✑✱ ✏s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✑ ❡ ✏s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✑✱ ❛❝♦♥s❡❧❤❛♠♦s q✉❡ ❧❡✐❛ ♦ ❝♦♥t❡ú❞♦ ❞♦ ❆♣ê♥❞✐❝❡ ❆ ❛♥t❡s ❞❡ ❝♦♥t✐♥✉❛r ❛ ❧❡✐t✉r❛ ❞❡st❡ ❝❛♣ít✉❧♦✳

Pr♦♣♦s✐çã♦ ✶✳✶ P❛r❛ t♦❞♦ m, n∈N✱

1 + 1 n

n <

1 + 1 m

m+1

✳

❉❡♠♦♥str❛çã♦✿ ❆ ✐♥❡q✉❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ❞❡♠♦♥str❛❞❛ ✉t✐❧✐③❛♥❞♦✲ s❡ ❛ ❝♦♥❤❡❝✐❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s ❛r✐t♠ét✐❝❛ ❡ ❣❡♦♠étr✐❝❛✳ ❙❛❜❡♠♦s q✉❡

n

√

a1a2· · ·an≤

a1+a2+· · ·+an

n ,

♦♥❞❡ a1, a2, . . . , an ∈R+✳ ❆❧é♠ ❞✐ss♦✱

n

√

a1a2· · ·an=

a1+a2+· · ·+an

n ⇔a1 =a2 =· · ·=an.

❉❡st❛ ❢♦r♠❛✱ ❝♦♠♦ 1₋ 1

m+1

<1< 1 + 1

n

♣❛r❛ t♦❞♦ t♦❞♦ m, n_∈N✱ ♦❜t❡♠♦s

m+n+1

s

1 + 1 n

n

1₋ 1 m+ 1

m+1

< n 1 +

1

n

+ (m+ 1) 1− 1

m+1

m+n+ 1 = 1

❡✱ ♣♦rt❛♥t♦✱

1 + 1 n

n

1₋ 1 m+ 1

m+1

<1,_∀m, n_∈N.

▲♦❣♦✱

1 + 1 n

n <

1 + 1 m

m+1

,_∀m, n_∈N.

Pr♦♣♦s✐çã♦ ✶✳✷ ❆ s❡q✉ê♥❝✐❛ (xn)n_∈N✱ ❞❡ t❡r♠♦ ❣❡r❛❧ xn = 1 + _n1 n

✱ é ♠♦♥ót♦♥❛ ❝r❡s❝❡♥t❡✳

❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✱ t❡♠♦s

1 + 1 n(n+ 2)

n(n+2)

<

1 + 1 n+ 1

n+2

.

P♦rt❛♥t♦✱

1 + 1 n(n+ 2)

n <

1 + 1 n+ 1

.

✶✳✸✳ ❖ ♥ú♠❡r♦ ❡ ❝♦♠♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛

❆ss✐♠✱ ❝♦♠♦

1 + 1 n(n+ 2)

n =

n(n+ 2) + 1 n(n+ 2)

n =

n2 _{+ 2n+ 1}

n(n+ 2)

n

= (n+ 1)

2n

nn_{(n+ 2)}n,

♦❜t❡♠♦s

(n+ 1)2n nn_{(n+ 2)}n <

1 + 1 n+ 1

= n+ 2 n+ 1.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

n+ 1 n

n <

n+ 2 n+ 1

n+1

.

▲♦❣♦✱

xn < xn+1.

P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✱ t❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦ ❣❡r❛❧ xn = 1 + 1

n

n

é ❧✐♠✐t❛❞❛✱ ❥á q✉❡✱ t♦♠❛♥❞♦✱ ♣♦r ❡①❡♠♣❧♦ m = 1✱

0<

1 + 1 n

n

<(1 + 1)2 = 4,_∀n_∈N.

❆ss✐♠✱ ❝♦♠♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✱ t❛❧ s❡q✉ê♥❝✐❛ t❛♠❜é♠ é ♠♦♥ót♦♥❛✱ ❝♦♥❝❧✉í♠♦s q✉❡(xn)n_∈Né ❝♦♥✈❡r❣❡♥t❡ ✭✈❡r ❚❡♦r❡♠❛ ❆✳✸ ❞♦ ❆♣ê♥❞✐❝❡ ❆✮✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡

✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ lim n→∞ 1 +

1

n

n

✳

❉❡✜♥✐çã♦ ✶✳✶ ❖ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r é ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ (xn)n∈N✱ ✐st♦ é✱

e= lim n→∞

1 + 1 n

n .

❊①✐st❡♠ ❞✐✈❡rs❛s ♦✉tr❛s ♠❛♥❡✐r❛s ❞❡ ❝❛r❛❝t❡r✐③❛r ♦ ♥ú♠❡r♦e✱ ❛♣❡s❛r ❞❛ ❞❡✜♥✐çã♦

✶✳✶ s❡r ❛ ♠❛✐s ✉s✉❛❧✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ♠♦str❛r❡♠♦s q✉❡ ♦e♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r

♠❡✐♦ ❞❛ sér✐❡P 1

n! ❡ q✉❡ t❛♠❜é♠ ❝♦rr❡s♣♦♥❞❡ à ❜❛s❡ ❞♦ ❧♦❣❛r✐t♠♦ q✉❡ ❞❡♥♦♠✐♥❛♠♦s

❞❡ ♥❛t✉r❛❧✳

✶✳✸ ❖ ♥ú♠❡r♦ ❡ ❝♦♠♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛

Pr♦♣♦s✐çã♦ ✶✳✸ ❖ ♥ú♠❡r♦ e é ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ (sn)n∈N ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é

❞❛❞♦ ♣♦r

sn= 1 + 1 1! +

1

2! +· · ·+ 1 n!,

✶✳✸✳ ❖ ♥ú♠❡r♦ ❡ ❝♦♠♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛ ✐st♦ é✱ e= ∞ X n=0 1 n!.

❉❡♠♦♥str❛çã♦✿ ❯t✐❧✐③❛♥❞♦ ❛ ❡①♣r❡ssã♦ ❞❡ ❡①♣❛♥sã♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ t❡♠♦s

1 + 1 n

n

= 1 +n_· 1 n +

n(n₋1) 2! ·

1

n2 +· · ·+

n(n₋1)_{· · ·}(n₋k+ 1)

k! ·

1

nk +· · ·+ 1 nn

= 1 + 1 + 1 2!

1₋ 1 n

+_{· · ·}+ 1 k!

1₋ 1

n 1−

2 n

· · ·

1₋ k−1 n +_{· · ·} + 1 n!

1₋ 1

n 1−

2 n

· · ·

1₋ n−1 n

.

❈♦♠♦ ❝❛❞❛ ❢❛t♦r ❡♥tr❡ ♣❛rê♥t❡s❡s ♥♦ ♠❡♠❜r♦ ❞✐r❡✐t♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ♥ã♦✲ ♥❡❣❛t✐✈♦ ❡ ♠❡♥♦r ❞♦ q✉❡ 1✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡

1 + 1 n

n

≤1 + 1 + 1

2!+· · ·+ 1

n! =sn,∀n ∈N.

❆❧é♠ ❞✐ss♦✱ ✜①❛♥❞♦ k_≥1 ❛r❜✐trár✐♦✱ t❡♠♦s q✉❡✱ s❡ n > k ❡♥tã♦

1 + 1 n

n

>1 + 1 + 1 2!

1− 1

n

+· · ·+ 1 k!

1− 1

n 1−

2 n

· · ·

1− k−1

n

.

❆ss✐♠✱ ❢❛③❡♥❞♦ n → ∞✱ ♦❜t❡♠♦s

e_≥1 + 1 + 1

2! +· · ·+ 1

k! =sk,∀k ∈N.

P♦rt❛♥t♦✱

1 + 1 n

n

≤sn_≤e,_∀n_∈N.

▲♦❣♦✱

e= lim n→∞

1 + 1 n

n

≤ lim

n→∞sn ≤e. ❈♦♥❝❧✉í♠♦s✱ ❞❡st❛ ❢♦r♠❛✱ q✉❡ lim

n→∞sn=e✳

❆ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛❞❛ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸ ♥♦s ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r✱ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡✱ ❛♣r♦①✐♠❛çõ❡s ❞❡ e✳

❊①❡♠♣❧♦✿ P❛r❛ n = 10✱ t❡♠♦s q✉❡

10

X

n=0

1

n! = 2,71828180114638447971781305,

q✉❡ ❥á é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❡ e ❝♦♠ ♣r❡❝✐sã♦ ❞❡ ✼ ❝❛s❛s ❞❡❝✐♠❛✐s✳

✶✳✸✳ ❖ ♥ú♠❡r♦ ❡ ❝♦♠♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛

✶✳✸✳✶ ■rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ ❡

❆ s❡❣✉✐r✱ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✹✱ é ❛♣r❡s❡♥t❛❞♦ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛ ❞❡♠♦♥str❛r ❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r✳

Pr♦♣♦s✐çã♦ ✶✳✹ P❛r❛ t♦❞♦ n_∈N✱

e=sn+ θn

n·n!✱ ❝♦♠ 0< θn<1.

❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸✱ ❞❛❞♦ n∈N✱ t❡♠♦s

e = 1 + 1 1!+

1

2!+· · ·+ 1 n!+

1 (n+ 1)! +

1

(n+ 2)! +· · ·

= sn+ 1 (n+ 1)! +

1

(n+ 2)! +· · · .

▲♦❣♦✱

e₋sn =

1 (n+ 1)! +

1

(n+ 2)! +· · ·

= 1

(n+ 1)!

1 + 1 n+ 2 +

1

(n+ 2)(n+ 3) +· · ·

< 1 (n+ 1)!

1 + 1 n+ 2 +

1

(n+ 2)2 +· · ·

= 1

(n+ 1)! · 1 1− 1

n+2

= n+ 2 (n+ 1)2_·n!

= n+ 2

(n2_{+ 2n+ 1)·n!}

= n+ 2

[n(n+ 2) + 1]_·n!

= n+ 2

[(n+ 2)(n+_n+21 )]·n!

= 1

(n+ _n+21 )·n!

< 1 n_·n!.

P♦rt❛♥t♦✱ ♣❛r❛ ❝❛❞❛ n∈N✱ ❡①✐st❡ θn ∈R✱ ❝♦♠ 0< θn<1✱ t❛❧ q✉❡

e=sn+ θn n_·n!.

✶✳✹✳ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

Pr♦♣♦s✐çã♦ ✶✳✺ ❖ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r é ✐rr❛❝✐♦♥❛❧✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ e s❡❥❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✱ ♦✉

s❡❥❛✱ q✉❡ ❡①✐st❡♠ p, q _∈Nt❛✐s q✉❡

e= p q.

❆ ♣❛rt✐r ❞❡ss❛ s✉♣♦s✐çã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ q!·e é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❆ss✐♠✱ ❞❡

❛❝♦r❞♦ ❝♦♠ ❛ Pr♦♣♦s✐çã♦ ✶✳✹✱ ❢❛③❡♥❞♦ n =q ♦❜t❡♠♦s

q!·e = q!·

sq+ θq q·q!

= q!·

1 + 1 + 1

2!+· · ·+ 1 q!+

θq q·q!

= q! +q! +q!

2 +· · ·+ q! q! +

θq q .

❈♦♠♦ q! +q! + q_2!!+· · ·+q_q!_! ∈N✱ t❡♠♦s q✉❡

θq

q =q!·e−

q! +q! +q_2!! +_{· · ·}+ q_q!_!∈N,

♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s 0 < θq

q < 1✳ ▲♦❣♦✱ ♥♦ss❛ s✉♣♦s✐çã♦ ♥ã♦ ♣♦❞❡ s❡r

✈❡r❞❛❞❡✐r❛ ❡✱ ♣♦rt❛♥t♦✱ e_∈R₋Q✳

❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞♦ ♥ú♠❡r♦ e ❛q✉✐ ❛♣r❡s❡♥t❛❞❛ ♣♦❞❡ s❡r ❡♥✲

❝♦♥tr❛❞❛ ♥♦ tr❛❜❛❧❤♦ ❞❡ ❑✉③✬♠✐♥ ❡ ❙❤✐rs❤♦✈ ✭✶✾✾✾✱ ♣✳✶✶✸✮✳

✶✳✹ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

◆❡st❛ s❡çã♦✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ sér✐❡ ❞❡ r❡s✉❧t❛❞♦s q✉❡ ♥♦s ❝♦♥❞✉③❡♠ à ❞❡♠♦♥s✲ tr❛çã♦ ❞❡ q✉❡ ♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r é ❛ ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s✳

❉❡✜♥✐çã♦ ✶✳✷ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ h:R∗_{+ →}R ❞❡✜♥✐❞❛ ♣♦r h(x) = 1/x✳ ❙❡❥❛ H ♦

❣rá✜❝♦ ❞❡ h✱ ✐st♦ é✱

H=

x,1 x

;x >0

.

❉❛❞♦s a, b∈R∗₊✱ ❞❡♥♦♠✐♥❛♠♦s ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡ ♦ ❝♦♥❥✉♥t♦ Hb

a ❞♦ ♣❧❛♥♦ ❧✐♠✐t❛❞♦

♣❡❧❛s r❡t❛s ✈❡rt✐❝❛✐s x=a ❡ x=b✱ ♣❡❧♦ ❡✐①♦ ❞❛s ❛❜❝✐ss❛s ❡ ♣❡❧❛ ❤✐♣ér❜♦❧❡ H✳

Pr♦♣♦s✐çã♦ ✶✳✻ P❛r❛ t♦❞♦ k >0✱ ❛s ❢❛✐①❛s Hb

a ❡ Hakbk tê♠ ❛ ♠❡s♠❛ ár❡❛✳

✶✳✹✳ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ a < b ❡ s❡❥❛ a = x0 <

x1 <· · ·< xi−1 < xi <· · ·< xn =b✱ ✉♠❛ ♣❛rt✐çã♦ ❞♦ ✐♥t❡r✈❛❧♦ [a, b]✱ ♥❛ q✉❛❧ t♦❞♦s

♦s s✉❜✐♥t❡r✈❛❧♦s [xi₋1, xi] ♣♦ss✉❡♠ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ✐❣✉❛❧ ❛ b−_na✳ ❚❡♠♦s q✉❡

n

X

i=1

b−a

nxi ≤ár❡❛ H b a≤

n

X

i=1

b−a nxi₋1

.

❉❛❞♦ k > 0✱ ❝♦♥s✐❞❡r❡ ❛ ♣❛rt✐çã♦ ak = x0k < x1k < · · · < xi−1k < xik < · · · < xnk = bk✱ ♥❛ q✉❛❧ ❝❛❞❛ s✉❜✐♥t❡r✈❛❧♦ [xi₋1k, xik] t❡♠ ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛❧ ❛ k(b_n−a)✳

❚❡♠♦s q✉❡

n

X

i=1

k(b−a)

nxik ≤ár❡❛ H bk ak ≤

n

X

i=1

k(b−a) nxi₋1k

.

❆ss✐♠✱ ❛ ♣❛rt✐r ❞❛s ✐♥❡q✉❛çõ❡s ❛♥t❡r✐♦r❡s✱ ♦❜t❡♠♦s

n

X

i=1

(b−a)

nxi −

n

X

i=1

(b−a)

nxi₋1 ≤ár❡❛

H_ab−ár❡❛ H_akbk ≤

n

X

i=1

(b−a) nxi₋1 −

n

X

i=1

(b−a) nxi .

▲♦❣♦✱

−(b−a)

n n X i=1 1 xi₋1 −

1 xi

≤ár❡❛ Hab−ár❡❛ Hakbk ≤

(b₋a) n n X i=1 1 xi₋1 −

1 xi . ❈♦♠♦ _n X i=1 1 xi₋1 −

1 xi

= 1 x0 −

1 xn = 1 a − 1 b =

(b−a) ab ,

t❡♠♦s q✉❡

−(b−a)

2

(ab)n ≤ár❡❛ H b

a−ár❡❛ Hakbk ≤

(b₋a)2

(ab)n .

❋❛③❡♥❞♦ n_{→ ∞}✱ ♦❜t❡♠♦s

0≤ár❡❛ H_ab−ár❡❛ H_akbk ≤0.

P♦rt❛♥t♦✱ ár❡❛ Hb

a =ár❡❛ Hakbk✳

P♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❛ ♥♦çã♦ ❞❡ ✏ár❡❛ ♦r✐❡♥t❛❞❛✑✱ ♦✉ s❡❥❛✱ ♣r♦✲ ✈✐❞❛ ❞❡ s✐♥❛❧ + ♦✉ −✱ ❡ ❝♦♥✈❡♥❝✐♦♥❛r❡♠♦s q✉❡ ❛ ár❡❛ ❞❛ ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡ s❡rá

♣♦s✐t✐✈❛ q✉❛♥❞♦ a < b✱ ♥❡❣❛t✐✈❛ q✉❛♥❞♦ b < a ❡ ③❡r♦ q✉❛♥❞♦ a=b✳ ❯t✐❧✐③❛r❡♠♦s ❛

♥♦t❛çã♦ ❛ s❡❣✉✐r✳

➪❘❊❆ Hb a=         

ár❡❛ Hb

a>0, s❡ a < b

−ár❡❛ Hb

a<0, s❡ b < a 0, s❡ a=b

.

✶✳✹✳ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

◆♦t❡ q✉❡ ➪❘❊❆ Hb

a =−➪❘❊❆ Hba✳

Pr♦♣♦s✐çã♦ ✶✳✼ ❉❛❞♦s a, b, c_∈R∗₊✱

➪❘❊❆ H_ab+➪❘❊❆ H_bc =➪❘❊❆ H_ac.

❉❡♠♦♥str❛çã♦✿ ❆♥❛❧✐s❛♥❞♦ ❝❛❞❛ ✉♠ ❞♦s s❡✐s ❝❛s♦s ♣♦ssí✈❡✐s✱ t❡♠♦s✿ ✭✐✮ ❙❡ a_≤b_≤c✱ ❡♥tã♦

➪❘❊❆ Hc

a =ár❡❛ Hac =ár❡❛ Hab+ár❡❛ Hbc =➪❘❊❆ Hab+➪❘❊❆ Hbc✳

✭✐✐✮ ❙❡ a_≤c_≤b✱ ❡♥tã♦

➪❘❊❆ Hb

a =ár❡❛ Hab =ár❡❛ Hac+ár❡❛ Hcb =➪❘❊❆ Hac−➪❘❊❆ Hbc✳

✭✐✐✐✮ ❙❡ b≤a≤c✱ ❡♥tã♦

➪❘❊❆ Hc

b =ár❡❛ Hbc =ár❡❛ Hba+ár❡❛ Hac =−➪❘❊❆ Hab +➪❘❊❆ Hac✳

✭✐✈✮ ❙❡ b_≤c_≤a✱ ❡♥tã♦

−➪❘❊❆ Hb

a=ár❡❛ Hba=ár❡❛ Hbc+ár❡❛ Hca=➪❘❊❆ Hbc−➪❘❊❆ Hac✳

✭✈✮ ❙❡ c≤a≤b✱ ❡♥tã♦

−➪❘❊❆ Hc

b =ár❡❛ Hcb =ár❡❛ Hca+ár❡❛ Hab =−➪❘❊❆ Hac +➪❘❊❆ Hab✳

✭✈✐✮ ❙❡ c_≤b _≤a✱ ❡♥tã♦

−➪❘❊❆ Hc

a=ár❡❛ Hca=ár❡❛ Hcb+ár❡❛ Hba=−➪❘❊❆ Hbc−➪❘❊❆ Hab✳

P♦rt❛♥t♦✱ ❞❛❞♦s a, b, c∈R∗₊✱

➪❘❊❆ Hab+➪❘❊❆ Hbc =➪❘❊❆ Hac✳

❉❡✜♥❛♠♦s ✉♠❛ ❢✉♥çã♦f :R∗_+→R✱ ♣♦♥❞♦✱ ♣❛r❛ ❝❛❞❛ x∈R∗₊✱

f(x) =➪❘❊❆ H₁x.

❈❧❛r❛♠❡♥t❡✱ f é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡✱ ♣♦rt❛♥t♦✱ ♠♦♥ót♦♥❛ ✐♥❥❡t✐✈❛✳ ❆❧é♠

❞✐ss♦✱ ♣❛r❛ ❝❛❞❛ x, y _∈R∗₊✱

f(xy) = ➪❘❊❆ H₁xy =➪❘❊❆ Hx

1 +➪❘❊❆ Hxxy.

✶✳✹✳ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

❈♦♠♦ ➪❘❊❆Hxy

x =➪❘❊❆ H

y

1✱ ♦❜t❡♠♦s

f(xy) =➪❘❊❆ H₁x+➪❘❊❆ H₁y =f(x) +f(y).

❉❡st❛ ❢♦r♠❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❧♦❣❛rít♠✐❝❛s ✭✈❡r ❆♣ê♥✲ ❞✐❝❡ ❇✮✱ ❡①✐st❡ a > 0✱ t❛❧ q✉❡ f(x) = log_ax ♣❛r❛ t♦❞♦ x ∈ R∗₊✳ ❊s❝r❡✈❡r❡♠♦s lnx

❡♠ ✈❡③ ❞❡ logax ❡ ❝❤❛♠❛r❡♠♦s ♦ ♥ú♠❡r♦ lnx ❞❡ ▲♦❣❛r✐t♠♦ ◆❛t✉r❛❧ ❞❡ x✳

❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱ t❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ f(x) = lnx é ✉♠❛ ❢✉♥çã♦ ❝r❡s✲

❝❡♥t❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ q✉❡ ❢♦✐ ❝♦♠❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♠❛✐♦r ❞♦ q✉❡ ✶✱ ♦✉ s❡❥❛✱ a > 1 ✭♣♦✐s f(a) = 1 > 0 = f(1)✮✳ Pr♦✈❛r❡♠♦s ❛ s❡❣✉✐r q✉❡ ❡ss❛ ❜❛s❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ◆ú♠❡r♦

❞❡ ❊✉❧❡r✳

Pr♦♣♦s✐çã♦ ✶✳✽ ❖ ♥ú♠❡r♦ e é ❛ ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ❛ ❋✐❣✉r❛ ✶✳✷✳

❋✐❣✉r❛ ✶✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢❛✐①❛ ❞❛ ❤✐♣ér❜♦❧❡ H₁1+x ♣❛r❛ ❡st✐♠❛t✐✈❛ ❞❡ ln(1 +x)✳

◆❡❧❛✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ✉♠ r❡tâ♥❣✉❧♦ ♠❡♥♦r✱ ❝♦♠ ❜❛s❡ ❡ ❛❧t✉r❛ ❞❡ ♠❡❞✐❞❛sx

❡ 1

1+x r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❢❛✐①❛H

1+x

1 ✱ ❡ ✉♠ r❡tâ♥❣✉❧♦ ♠❛✐♦r✱ ❝✉❥❛ ❜❛s❡ ♠❡❞❡x❡ ❝✉❥❛

❛❧t✉r❛ é ✐❣✉❛❧ ❛ 1✳ ❈♦♠♣❛r❛♥❞♦ ❛s ár❡❛s ❞❡ss❛s três r❡❣✐õ❡s ❞♦ ♣❧❛♥♦✱ t❡♠♦s q✉❡✱

♣❛r❛ t♦❞♦ x >0✱

x

1 +x <ln(1 +x)< x.

❉✐✈✐❞✐♥❞♦ ❝❛❞❛ ♠❡♠❜r♦ ❞❡st❛ ✐♥❡q✉❛çã♦ ♣♦r x✱ ♦❜t❡♠♦s

1 1 +x <

ln(1 +x) x <1.

✶✳✹✳ ❖ ♥ú♠❡r♦ ❡ ❡ ♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s

❆ss✐♠✱ ❞❛❞♦ n _∈N✱ s❡ t♦♠❛r♠♦sx= 1

n✱ t❡♠♦s n

n+ 1 <ln

1 + 1 n

n <1.

▲♦❣♦✱ ❝♦♠♦ lnk = log_ak✱ ❝♦♠ a >1✱

ann+1 _<

1 + 1 n

n < a.

❈♦♠♦ n

n+1 s❡ ❛♣r♦①✐♠❛ ❞❡ 1 q✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❢❛③❡♥❞♦ n → ∞✱

♦❜t❡♠♦s

a_≤ lim n→∞

1 + 1 n

n

≤a,

❡✱ ♣♦rt❛♥t♦✱

a= lim n→∞

1 + 1 n

n =e.

❆ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❊✉❧❡r ❝♦♠♦ ❛ ❜❛s❡ ❞♦s ▲♦❣❛r✐t♠♦s ◆❛t✉r❛✐s ♥♦s ♣❡r♠✐t❡ ♣r♦✈❛r ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦✿ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ e é ✐❣✉❛❧ à

s✉❛ ♣ró♣r✐❛ ❞❡r✐✈❛❞❛✳

✶✳✹✳✶ ❯♠❛ ❝♦♥s✐❞❡r❛çã♦ s♦❜r❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡

❡

◆❡st❛ s✉❜s❡çã♦✱ ✉t✐❧✐③❛♠♦s ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ▲♦❣❛r✐t♠♦ ◆❛t✉r❛❧ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ❛ ❢✉♥çã♦ f(x) =ex _{❡ s✉❛ ❞❡r✐✈❛❞❛ sã♦ ✐❣✉❛✐s✳}

Pr♦♣♦s✐çã♦ ✶✳✾ ❙❡❥❛ f :R_→R∗₊ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f(x) =ex_{✳ ❆ ❞❡r✐✈❛❞❛}

❞❡ f é ❞❛❞❛ ♣♦r

f′(x) = ex.

❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡

f′(x) = lim h→0

f(x+h)₋f(x) h

= lim h→0

ex+h_−ex h

= lim h→0

ex_·eh_−ex h

= lim h→0e

x

· e

h₋₁

h

= ex_·lim h→0

eh₋₁

h .