❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦
♣♦r ▼❡✐♦ ❞❡ ❙❡q✉ê♥❝✐❛s ❡ ❙ér✐❡s ◆✉♠ér✐❝❛s
❈❧❛✉❞✐♦ ❋❡r♥❛♥❞❡s ❱❛s❝♦♥❝❡❧♦s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s
Vasconcelos, Claudio Fernandes
Modelagem matemática no ensino médio por meio de seqüências e séries numéricas / Claudio Fernandes Vasconcelos. - Rio Claro, 2016 63 f. g il., figs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientadorg Ricardo de Sá Teles
1. Matemática. 2. Modelagem matemática. 3. Seqüências numéricas. 4. Series numéricas. I. Título.
510 V331m
Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESP Campus de Rio Claro/SP
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❈❧❛✉❞✐♦ ❋❡r♥❛♥❞❡s ❱❛s❝♦♥❝❡❧♦s
▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣♦r ▼❡✐♦ ❞❡
❙❡q✉ê♥❝✐❛s ❡ ❙ér✐❡s ◆✉♠ér✐❝❛s
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s ❖r✐❡♥t❛❞♦r
Pr♦❢✭❛✮✳ ❉r✭❛✮✳ ➱r✐❝❛ ❘❡❣✐♥❛ ❋✐❧❧❡tt✐ ◆❛s❝✐♠❡♥t♦ ❉❋◗ ✲ ❯◆❊❙P ❆❘❆❘❆◗❯❆❘❆
Pr♦❢✭❛✮✳ ❊r✐❦❛ ❈❛♣❡❧❛t♦
❋❈▲❆❘ ✲ ❯◆❊❙P ❆❘❆❘❆◗❯❆❘❆
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❞✐s❝♦rr❡♠♦s ❜r❡✈❡♠❡♥t❡ ❛ r❡s♣❡✐t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡♥s✐♥♦ ❡ ♣❡sq✉✐s❛ ❡ ❞♦ ♠♦❞♦ ❝♦♠♦ ❡❧❛ é ❛❜♦r❞❛❞❛ ♥♦s ♣❛râ♠❡tr♦s ❝✉r✲ r✐❝✉❧❛r❡s ♥❛❝✐♦♥❛✐s✳ ❆♣r❡s❡♥t❛♠♦s ✈ár✐♦s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s ❡✱ ♣♦r ✜♠✱ ❝♦❧♦❝❛♠♦s ❛❧❣✉♠❛s ♣r♦♣♦st❛s ♣❡❞❛❣ó❣✐❝❛s ✉t✐❧✐③❛♥❞♦ ❛ ♠♦❞❡❧❛✲ ❣❡♠ ♠❛t❡♠át✐❝❛ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ t❡♦r✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s✳ ❖ ✐♥t✉✐t♦ ❞❡st❛s ♣r♦♣♦st❛s é ♣r♦♣♦r❝✐♦♥❛r ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ♥♦s ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳
❆❜str❛❝t
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❊t❛♣❛s ❞❛ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ❡ ❛s ❆çõ❡s ❈♦❣♥✐t✐✈❛s ❞♦s ❆❧✉♥♦s ❬❉❊ ❆▲▼❊■❉❆✱ ▲✳ ▼ ❲✳❀ ❉❆ ❙■▲❱❆✱ ❑✳ P✳ ❈✐ê♥❝✐❛&❊❞✉❝❛çã♦✱ ✈✳ ✶✽✱
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s ✷✸
✷✳✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ▲✐♠✐t❛çã♦ ❡ ▼♦♥♦t♦♥✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹ ▲✐♠✐t❡s ■♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸ ❙ér✐❡s ◆✉♠ér✐❝❛s ✸✺
✸✳✶ ❋✉♥❞❛♠❡♥t♦s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❙ér✐❡s ❞❡ ❚❡r♠♦s P♦s✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✸ ❙ér✐❡s ❆❧t❡r♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹ ❊st✐♠❛t✐✈❛ ❞♦ ❊rr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✺ ❖ ❚❡st❡ ❞❛ ❘❛③ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❙❡q✉ê♥❝✐❛s ❡ ❙ér✐❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✺✼ ✹✳✶ Pr♦♣♦st❛s ❞❡ ❆t✐✈✐❞❛❞❡s ❉✐❞át✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✸
✶ ■♥tr♦❞✉çã♦
◆❡st❡ ❝❛♣ít✉❧♦ ❞✐s❝♦rr❡♠♦s ❜r❡✈❡♠❡♥t❡ ❛ r❡s♣❡✐t♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❞❡s✲ t❛❝❛♥❞♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ✐♥tr♦❞✉③✐✲❧❛ ♥♦ ❛♠❜✐❡♥t❡ ❞❛ s❛❧❛ ❞❡ ❛✉❧❛✳
❆♦ ❧♦♥❣♦ ❞♦s ❛♥♦s ♥❛ ❧✐t❡r❛t✉r❛ ❢♦r❛♠ ❡s❝r✐t❛s ❞✐✈❡rs❛s ❞❡✜♥✐çõ❡s ♣❛r❛ ♦ t❡r♠♦ ♠♦✲ ❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ❇❛ss❛♥❡③✐ ❛ss✉♠❡ q✉❡ ✏✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ sí♠❜♦❧♦s ❡ r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ r❡♣r❡s❡♥t❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♦ ♦❜❥❡t♦ ❡st✉❞❛❞♦✑ ✭✷✵✵✷✱ ♣✳✷✵✮✳
◗✉❛♥❞♦ ❢❛❧❛♠♦s ❡♠ ♠♦❞❡❧❛❣❡♠ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❡❞✉❝❛❝✐♦♥❛❧ ♣♦❞❡♠♦s ❞❡✜♥✐✲❧♦ ❝♦♠♦ ✉♠ ❛♠❜✐❡♥t❡ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ♥♦ q✉❛❧ ♦s ❛❧✉♥♦s sã♦ ❝♦♥✈✐❞❛❞♦s ❛ ✐♥❞❛❣❛r ❡✴♦✉ ✐♥✈❡st✐❣❛r✱ ♣♦r ♠❡✐♦ ❞❛ ♠❛t❡♠át✐❝❛✱ s✐t✉❛çõ❡s ♦r✐✉♥❞❛s ❞❡ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❬▼❖◆❚❊■❘❖✱ ✶✾✾✶❪✱ ❡①✐st❡♠ ❞✉❛s ✈❡rt❡♥t❡s q✉❡ ✉t✐❧✐③❛♠ ❛ ▼♦❞❡✲ ❧❛❣❡♠✿ ♦s q✉❡ ❛ ✈❡❡♠ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❞❡ ♣❡sq✉✐s❛ ❡♠ ▼❛t❡♠át✐❝❛ ❡ ♦s q✉❡ ❛ ✈❡❡♠ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ♣❡❞❛❣ó❣✐❝♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ▼❛t❡♠át✐❝❛✳
P♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❡①✐st❡♠ ❡st✉❞♦s s♦❜r❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥❛ ár❡❛ ❞♦ ❡♥s✐♥♦ ✉t✐❧✐③❛♥❞♦✲❛ ❝♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ♣❡❞❛❣ó❣✐❝❛ ❡ ♥❛ ár❡❛ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ♣❡sq✉✐s❛ ❛❝❛❞ê♠✐❝❛✳ ❖❜s❡r✈❛♥❞♦ q✉❡ ✉♠ ❣r✉♣♦ ♥ã♦ ❡①✲ ❝❧✉✐ ♦ ♦✉tr♦✱ ♦s ♠♦❞❡❧♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ♠❡✐♦ ❞❡ ♣❡sq✉✐s❛s ❛❝❛❞ê♠✐❝❛s✱ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ q✉❛♥❞♦ ❞❡✈✐❞❛♠❡♥t❡ ❛❞❛♣t❛❞♦s
P❛r❛ ♦ ❣r✉♣♦ q✉❡ ❞❡s❡♥✈♦❧✈❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ♣❡sq✉✐s❛✱ ❛ ♠♦❞❡❧❛❣❡♠ é ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❛ ♣❛rt✐r ❞❡ ✉♠ ❢❛t♦ ❞❛ r❡❛❧✐❞❛❞❡ ♦♥❞❡ sã♦ ❧❡✈❛♥t❛❞❛s ❤✐♣ót❡s❡s✱ ❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ♣♦r ♠❡✐♦ ❞❡ ❝á❧❝✉❧♦s ♠❛t❡♠át✐❝♦s ❡ ♣♦s✲ t❡r✐♦r♠❡♥t❡ ✈❡r✐✜❝❛✲s❡ ❛ s✉❛ ✈❛❧✐❞❛❞❡✳ ❈❛s♦ s❡❥❛ ❝♦♥st❛t❛❞♦ q✉❡ ❛ s♦❧✉çã♦ ♥ã♦ é ✈á❧✐❞❛✱ ♥♦✈❛s ❤✐♣ót❡s❡s sã♦ ❡❧❛❜♦r❛❞❛s ❡ ♦ ♣r♦❝❡ss♦ r❡❝♦♠❡ç❛✳ ❊st❡ ❣r✉♣♦ r❡❛❧✐③❛ ♣❡sq✉✐s❛s ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ P✉r❛✱ ♦♥❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s sã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ r❡s♦❧✈❡r ✉♠ ❞❛❞♦ ♣r♦❜❧❡♠❛ ❡ ❞❡ ♣♦ss❡ ❞❡st❡s ♠♦❞❡❧♦s✱ ♦s ♣❡sq✉✐s❛❞♦r❡s ♣♦❞❡♠ r❡❛❧✐③❛r ♣r❡✈✐sõ❡s✱ ❡❧❛❜♦r❛r ❤✐♣ót❡s❡s ❡ ❝♦♥❥❡❝t✉r❛s s♦❜r❡ ♦ ❢❡♥ô♠❡♥♦ ❡st✉❞❛❞♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♦❜❥❡t✐✈♦ ❞♦ ❡st✉❞♦✱ ❞♦s ❞❛❞♦s ❡ ♠❛t❡r✐❛✐s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ✐ss♦✳
❏á ♣❛r❛ ♦ ❣r✉♣♦ q✉❡ ✉t✐❧✐③❛ ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡♥s✐♥♦✱ ❛ ▼♦❞❡❧❛✲ ❣❡♠ é ❡♥❝❛r❛❞❛ ❝♦♠♦ ✉♠ ❝❛♠✐♥❤♦ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✱ ♦❜s❡r✈❛♥❞♦ ❛ r❡❛❧✐❞❛❞❡ ❞♦ ❛❧✉♥♦ ❡✱ ✏❛ ♣❛rt✐r ❞❡ s❡✉s q✉❡st✐♦♥❛♠❡♥t♦s s❡ ❞❡❢r♦♥t❛ ❝♦♠ ♣r♦❜❧❡♠❛s q✉❡ ❞❡✈❡♠ ♠♦❞✐✜❝❛r t❛♥t♦ ❛ s✉❛ ❛çã♦✱ ❝♦♠♦ s✉❛ ❢♦r♠❛ ❞❡ ♦❜s❡r✈❛r t❛❧
✷✵ ■♥tr♦❞✉çã♦
♠✉♥❞♦✑ ❬▼❖◆❚❊■❘❖✱ ✶✾✾✶✱ ♣✳ ✶✵✻❪✳
❆ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦s ❧❡✈❛ ❛♦ ❡♥❝♦♥tr♦ ❞❡ s✐t✉❛çõ❡s ❡♠ q✉❡ t❡♠♦s ❢❡♥ô✲ ♠❡♥♦s ❞♦ ❝♦t✐❞✐❛♥♦ ♠♦❞❡❧❛❞♦s ♣❛r❛ s❡ t♦r♥❛r❡♠ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ ♣♦ss❛♠ s❡r t❡st❛❞♦s ❡ ✈❛❧✐❞❛❞♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ♥ã♦ sã♦ ♣♦✉❝♦s ♦s ♣❡sq✉✐s❛❞♦r❡s q✉❡ ❛ss♦❝✐❛♠ ❛ ❡✈♦❧✉çã♦ ❞❛ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠ ❛ ♣ró♣r✐❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ P❛r❛ ❇✐✲ ❡♠❜❡♥❣✉t ❡ ❍❡✐♥✱ ✏❛ ♠♦❞❡❧❛❣❡♠ é tã♦ ❛♥t✐❣❛ ❝♦♠♦ ❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛✱ s✉r❣✐♥❞♦ ❞❡ ❛♣❧✐❝❛çõ❡s ♥❛ r♦t✐♥❛ ❞✐ár✐❛ ❞♦s ♣♦✈♦s ❛♥t✐❣♦s✑ ❬❇■❊▼❇❊◆●❯❚✱ ▼✳ ❙✳ ❍❊■◆✱ ✷✵✵✷✱ ♣✳ ✽❪✳
P♦❞❡♠♦s ❝✐t❛r ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s r❡❛✐s q✉❡ ❢♦r❛♠ ♠♦❞❡❧❛❞♦s ❡ s❡ t♦r♥❛r❛♠ ♠♦❞❡❧♦s ❞❡ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✱ q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡✿
• ❖s ♠♦❞❡❧♦s ♣♦♣✉❧❛❝✐♦♥❛✐s ♠❛t❡♠át✐❝♦s✱ q✉❡ sã♦ ❞❡s❝r✐t♦s ❛♣ós ❛ ♦❜s❡r✈❛çã♦ ❞❡
❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s ❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢❛✉♥❛ ❡ ✢♦r❛✱ ♣♦❞❡♠ ✐♠♣❡❞✐r ❛ ❡①t✐♥✲ çã♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❡s♣é❝✐❡ ♦✉ ♠❡s♠♦ ♦ ❡s❣♦t❛♠❡♥t♦ ❞♦s r❡❝✉rs♦s ♥❛t✉r❛✐s ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r❡❣✐ã♦✳ ❉❡♥tr❡ ❡ss❡s ♠♦❞❡❧♦s ♣♦❞❡♠♦s ❞❡st❛❝❛r ❞♦✐s ❞♦s ♠❛✐s ❝♦♥❤❡❝✐❞♦s✳ ❖ ♣r✐♠❡✐r♦ é ❞♦ ❡❝♦♥♦♠✐st❛ ✐♥❣❧ês ❚❤♦♠❛s ▼❛❧t❤✉s✱ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✶✼✾✽✳ ❖ ♠♦❞❡❧♦ ♠❛❧t❤✉s✐❛♥♦ ♣r❡ss✉♣õ❡ q✉❡ ❛ t❛①❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ❛ ♣♦♣✉❧❛çã♦ ❞❡ ✉♠ ♣❛ís ❝r❡s❝❡ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ ♣♦♣✉❧❛çã♦ t♦t❛❧ ❞♦ ♣❛ís ♥❛q✉❡❧❡ ✐♥st❛♥t❡✿
dP
dt =kP,
♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✭♥❡ss❡ ❝❛s♦✮ k >0✳
❙❛❜❡♥❞♦✲s❡ q✉❡ ✉♠❛ ❝❡rt❛ ♣♦♣✉❧❛çã♦ ❝r❡s❝❡ s❡❣✉♥❞♦ ♦ ♠♦❞❡❧♦ ♠❛❧t❤✉s✐❛♥♦ ❡ ✱ s❛❜❡♥❞♦ ❛✐♥❞❛✱ q✉❡ P(0) =P0✱ ❡♥tã♦ t❡♠♦s✿
P =P0ekt.
❖ ♠♦❞❡❧♦ ❞✐s❝r❡t♦ ❞❡ ▼❛❧t❤✉s é ❞❛❞♦ ♣♦rP(t+ 1)−P(t) =αP(t)✳ ❙❡P(0) =P0✱
❝♦♠ α ❝♦♥st❛♥t❡ ❡ P(0) ❛ ♣♦♣✉❧❛çã♦ ♥♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧✱ t❡♠♦s P(t) = (1 +α)tP
0.
❖ s❡❣✉♥❞♦ ♠♦❞❡❧♦ é ♦ ❞❡ ❱❡r❤✉❧st q✉❡ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❜❡❧❣❛ q✉❡ ✐♥tr♦❞✉✲ ③✐✉ ❛ ❡q✉❛çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧♦❣íst✐❝♦ ♦♥❞❡ ❛ ♣♦♣✉❧❛çã♦ ❝r❡s❝❡ ❛té ✉♠ ❧✐♠✐t❡ ♠á①✐♠♦ s✉st❡♥tá✈❡❧✱ ✐st♦ é✱ ❡❧❛ t❡♥❞❡ ❛ s❡ ❡st❛❜✐❧✐③❛r✳ ❖ ♠♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st é✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦ ♠♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s ♠♦❞✐✜❝❛❞♦✿
dP dt =rP
1− P
P∞
P(0) =P0, r >0,
✷✶
• ❖ ♠♦❞❡❧♦ ❞♦s ❈♦r♣♦s ❋❧✉t✉❛♥t❡s ❞❡ ❆rq✉✐♠❡❞❡s✿ ❊♥tr❡ ♦s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s
♣♦r ❆rq✉✐♠❡❞❡s✱ ❡①✐st❡ ♦ tr❛t❛❞♦ ❙♦❜r❡ ♦s ❈♦r♣♦s ❋❧✉t✉❛♥t❡s✱ ♦♥❞❡ é ❡♥❝♦♥tr❛❞♦ ♦ q✉❡ ❤♦❥❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ❚❡♦r❡♠❛ ♦✉ Pr✐♥❝í♣✐♦ ❞❡ ❆rq✉✐♠❡❞❡s✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ❡❧❡ ❛✜r♠❛ q✉❡ ✏t♦❞♦ ❝♦r♣♦ ♠❡r❣✉❧❤❛❞♦ ❡♠ ✉♠ ✢✉✐❞♦ r❡❝❡❜❡ ✉♠ ❡♠♣✉①♦✱ ❞❡ ❜❛✐①♦ ♣❛r❛ ❝✐♠❛✱ ✐❣✉❛❧ ❛♦ ♣❡s♦ ❞♦ ✈♦❧✉♠❡ ❞♦ ✢✉✐❞♦ ❞❡s❧♦❝❛❞♦✑✳ ❍á r❡❧❛t♦s ❞❡ q✉❡ ♦ r❡✐ ❍✐❡r♦♥ ❞❡ ❙✐r❛❝✉s❛ ❞❡❝✐❞✐✉ ❝♦❧♦❝❛r ♥♦ t❡♠♣❧♦ ✉♠❛ ❝♦r♦❛ ❞❡ ♦✉r♦ ❝♦♠♦ ♦❢❡rt❛ ♣❛r❛ ♦s ❞❡✉s❡s✱ ♠❛s ♦ ♦✉r✐✈❡s ♠✐st✉r♦✉ ♣r❛t❛ ❝♦♠ ♦✉r♦ ♥❛ ❝♦♥❢❡❝çã♦ ❞❛ ♠❡s♠❛✳ ❉❡s❝♦♥✜❛❞♦ ♦ r❡✐ ♣r♦♣ôs q✉❡ ❆rq✉✐♠❡❞❡s r❡s♦❧✈❡ss❡ ♦ ♣r♦❜❧❡♠❛✳ ❆ s♦❧✉çã♦ ❢♦✐ ❡♥❝♦♥tr❛❞❛ ♣♦r ❆rq✉✐♠❡❞❡s ❛♦ ❜❛♥❤❛r✲s❡✳
• ❆ ❤✐stór✐❛ ❞❛ ♠❛çã ❞❡ ◆❡✇t♦♥ ❛♣❛r❡❝❡✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ❊❧❡♠❡♥t♦s ❞❛ ❋✐❧♦✲
s♦✜❛ ❞❡ ◆❡✇t♦♥✱ ❡s❝r✐t♦ ♣♦r ❱♦❧t❛✐r❡ ❡ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✼✸✽✳ ◆❡st❡ ❧✐✈r♦✱ ❱♦❧t❛✐r❡ ✲ q✉❡ ❛❞♠✐r❛✈❛ ♠✉✐t♦ ❙✐r ■s❛❛❝ ❡ s✉❛s t❡♦r✐❛s ✲ ❛♣r❡s❡♥t♦✉ ✉♠❛ ❝❧❛r❛ ❡ ❛❞♠✐rá✈❡❧ ✐♥t❡r♣r❡t❛çã♦ ❞❛s ✐❞❡✐❛s ♥❡✇t♦♥✐❛♥❛s✳ ❆ ❧❡♥❞❛ ❞❛ ♠❛çã ❢♦✐ ❡s♣❛❧❤❛❞❛ ♣❡❧❛ s♦❜r✐✲ ♥❤❛ ❞♦ ❝✐❡♥t✐st❛ ✐♥❣❧ês✱ ❈❛t❤❡r✐♥❡ ❇❛rt♦♥ ❈♦♥❞✉✐tt✱ ❡ s❡✉ ♠❛r✐❞♦✱ q✉❡ ✈✐✈❡r❛♠ ❝♦♠ ❡❧❡ ♥♦s ú❧t✐♠♦s ❛♥♦s ❞❡ ✈✐❞❛ ❞♦ ❝✐❡♥t✐st❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♣ró♣r✐♦ ◆❡✇t♦♥ ❝♦♥t♦✉ ❛♦ ❡st✉❞✐♦s♦ ❲✐❧❧✐❛♠ ❙t✉❦❡❧❡② t❡r s✐❞♦ ✐♥s♣✐r❛❞♦ ♣♦r ✉♠❛ ♠❛çã ❝❛✐♥❞♦ ❡♠ s❡✉ q✉✐♥t❛❧ ✲ ❡ ♥ã♦ ❡♠ s✉❛ ❝❛❜❡ç❛ ✲ ❛♦ ✐♥✈❡st✐❣❛r ❛ t❡♦r✐❛ ❞❛ ❣r❛✈✐t❛çã♦✳ ❙t✉❦❡❧❡② r❡❧❛t❛ ❛ ❝♦♥✈❡rs❛ q✉❡ t❡✈❡ ❝♦♠ ◆❡✇t♦♥ ♥♦ ❧✐✈r♦ ▼❡♠ór✐❛ ❞❡ ❙✐r ■s❛❛❝ ◆❡✇t♦♥✱ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✼✺✷✳ ❊ss❡ ✐♥❝✐❞❡♥t❡ ❝♦t✐❞✐❛♥♦ s❡r✈✐✉ ♣❛r❛ ✐♥s♣✐r❛r ✉♠❛ t❡♦r✐❛ ❝❛♣❛③ ❞❡ ♠✉❞❛r ♦ ♠✉♥❞♦ ♣❛r❛ s❡♠♣r❡✳
• ◆♦ sé❝✉❧♦ ❱ ❛✳❈✳✱ ♦s ❡❣í♣❝✐♦s✱ s❡❣✉♥❞♦ ♦ ❣r❡❣♦ ❍❡ró❞♦t♦✱ ✉s❛✈❛♠ ❝♦♥❝❡✐t♦s ❞❡
❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ♣❛r❛ q✉❡✱ ❛♣ós ❛s ❡♥❝❤❡♥t❡s ❞♦ r✐♦ ◆✐❧♦✱ ♦s ❛❣r✐♠❡♥s♦r❡s ❞❡t❡r✲ ♠✐♥❛ss❡♠ ❛ r❡❞✉çã♦ s♦❢r✐❞❛ ♣❡❧♦ t❡rr❡♥♦✱ ♣❛ss❛♥❞♦ ♦ ♣r♦♣r✐❡tár✐♦ ❛ ♣❛❣❛r ✉♠ tr✐❜✉t♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ q✉❡ r❡st❛r❛✳
◆♦ ❇r❛s✐❧✱ ❛ ▼♦❞❡❧❛❣❡♠ ❡♠ ❊❞✉❝❛çã♦ ▼❛t❡♠át✐❝❛ ❝♦♠❡ç♦✉ ❛ ❡♠❡r❣✐r ❛ ♣❛rt✐r ❞❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ♦♥❞❡ ❡st❛✈❛ ♣r❡s❡♥t❡ ❡♠ ❛❧❣✉♠❛s ❞✐s❝✐♣❧✐♥❛s ❞❛s ✉♥✐✈❡rs✐❞❛❞❡s ❞♦s ❡st❛❞♦s ❞❡ ❙ã♦ P❛✉❧♦ ❡ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳
P❛r❛ ❯❜✐r❛t❛♥ ❉✬❆♠❜rós✐♦ ❡ ❘♦❞♥❡② ❈❛r❧♦s ❇❛ss❛♥❡③✐✱ ❛ ▼♦❞❡❧❛❣❡♠ s✉r❣✐✉ ✐♥s♣✐✲ r❛❞❛ ♥❛ ❡t♥♦♠❛t❡♠át✐❝❛✱ q✉❡ ❢♦✐ ❞❡✜♥✐❞❛ ♣♦r ❉✬❆♠❜rós✐♦ ❝♦♠♦ ✏❛ ♠❛t❡♠át✐❝❛ ✉s❛❞❛ ♣♦r ✉♠ ❣r✉♣♦ ❝✉❧t✉r❛❧ ❞❡✜♥✐❞♦ ♥❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡ ❛t✐✈✐❞❛❞❡s ❞♦ ❞✐❛ ❛ ❞✐❛✑ ❡ t✐♥❤❛ ❝♦♠♦ ♣r♦♣♦st❛ ❛❜♦r❞❛r ❛ ▼❛t❡♠át✐❝❛ ❛ ♣❛rt✐r ❞♦ ❝♦♥t❡①t♦ s♦❝✐❛❧ ❞♦s ❛❧✉♥♦s✳
❉✬❆♠❜rós✐♦ ❛❝r❡❞✐t❛ q✉❡ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ é ❜❛s❡❛❞♦ ♥♦ ❡st✉❞♦ ❞❡ ✉♠ ❢❡♥ô♠❡♥♦ r❡❛❧ ♣♦r ✉♠ ✐♥❞✐✈í❞✉♦ ♦♥❞❡ ❡st❡ ♦ tr❛❞✉③ ❡♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡✲ ♠át✐❝❛✱ ♣❛r❛ q✉❡ ❞❡ss❛ ❢♦r♠❛ ♣♦ss❛ ❡st✉❞á✲❧♦✳
◆♦ ❡♥s✐♥♦ tr❛❞✐❝✐♦♥❛❧ ❛t✉❛❧♠❡♥t❡ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❡s❝♦❧❛s ❞♦ ♣❛ís✱ ❛ ❛♣r❡♥❞✐③❛❣❡♠ é ❝❡♥tr❛❧✐③❛❞❛ ♥❛ ✜❣✉r❛ ❞♦ ♣r♦❢❡ss♦r ❡ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞❛ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ é ❝♦♠♣❛rt✐❧❤❛❞♦ ❝♦♠ ♦ ❣r✉♣♦ ❞❡ ❛❧✉♥♦s✳
✷✷ ■♥tr♦❞✉çã♦
• ▼♦t✐✈❛çã♦ ♣♦r ♣❛rt❡ ❞❡ ❡❞✉❝❛♥❞♦ ❡ ❡❞✉❝❛❞♦r✳
• ❋❛❝✐❧✐❞❛❞❡ ❞❡ ❛♣r❡♥❞❡r ✲ ♦ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦ ♣❛ss❛ ❞❡ ❛❜str❛t♦ ❛ ❝♦♥❝r❡t♦✳ • Pr❡♣❛r❛çã♦ ♣❛r❛ ❢✉t✉r❛s ♣r♦✜ssõ❡s ♥❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡✲
✈✐❞♦ ❛ ✐♥t❡r❛t✐✈✐❞❛❞❡ ❞❡ ❝♦♥t❡ú❞♦s✳
• ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦✳
• ❖♣♦rt✉♥✐❞❛❞❡ ❞♦ ❛❧✉♥♦ ❛ss✉♠✐r ♦ ♣❛♣❡❧ ❞❡ ❝✐❞❛❞ã♦ ❝rít✐❝♦ ❡ tr❛♥s❢♦r♠❛❞♦r ❞❡
s✉❛ r❡❛❧✐❞❛❞❡✳
• ❈♦♠♣r❡❡♥sã♦ ❞♦ ♣❛♣❡❧ s♦❝✐♦❝✉❧t✉r❛❧ ❞❛ ▼❛t❡♠át✐❝❛✱ t♦r♥❛♥❞♦✲❛ ❛ss✐♠ ♠❛✐s ✐♠✲
♣♦rt❛♥t❡✳
❆ ♦❜t❡♥çã♦ ❞❡ ♥♦✈❛s ❢♦r♠❛s ♣❛r❛ q✉❡ ❛ ♠❛t❡♠át✐❝❛ s❡❥❛ ❡♥s✐♥❛❞❛ ❛tr❛✈és ❞❛ ❝♦♠✲ ♣r❡❡♥sã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❡ s✐t✉❛çõ❡s r❡❛✐s ❝♦♠ ❢❡rr❛♠❡♥t❛❧ ♠❛t❡♠át✐❝♦ ♣♦❞❡ s❡r ❝♦♠♣♦st♦ ♣♦r ❡t❛♣❛s✳ ❬❇■❊▼❇❊◆●❯❚✱ ▼✳ ❙✳ ❍❊■◆✱ ✷✵✵✷❪ ❞❡st❛❝❛ ❛s s❡❣✉✐♥✲ t❡s✿
✶✲ ■♥t❡r❛çã♦ ✲ ♦♥❞❡ ♦❝♦rr❡ ♦ ❡♥✈♦❧✈✐♠❡♥t♦ ❝♦♠ ♦ t❡♠❛ ✭r❡❛❧✐❞❛❞❡✮ ❛ s❡r ❡st✉❞❛❞♦ ❡ ♣r♦❜❧❡♠❛t✐③❛❞♦✱ ❛tr❛✈és ❞❡ ✉♠ ❡st✉❞♦ ✐♥❞✐r❡t♦ ✭♣♦r ♠❡✐♦ ❞❡ ❥♦r♥❛✐s✱ ❧✐✈r♦s ❡✴♦✉ r❡✈✐st❛s✮ ♦✉ ❞✐r❡t♦ ✭♣♦r ♠❡✐♦ ❞❡ ❡①♣❡r✐ê♥❝✐❛s ❡♠ ❝❛♠♣♦✮✳
✷✲ ▼❛t❡♠❛t✐③❛çã♦ ✲ ♦♥❞❡ ♦❝♦rr❡ ❛ ✏tr❛❞✉çã♦✑ ❞❛ s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ➱ ❛q✉✐ q✉❡ s❡ ❢♦r♠✉❧❛ ✉♠ ♣r♦❜❧❡♠❛ ❡ ❡s❝r❡✈❡✲♦ s❡❣✉♥❞♦ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ ❧❡✈❡ à s♦❧✉çã♦✳
✸✲ ▼♦❞❡❧♦ ▼❛t❡♠át✐❝♦ ✲ ♦♥❞❡ ♦❝♦rr❡ ❛ ✏t❡st❛❣❡♠✑ ♦✉ ✈❛❧✐❞❛çã♦ ❞♦ ♠♦❞❡❧♦ ♦❜t✐❞♦✱ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❛s r❡s♣♦st❛s q✉❡ ♦ ♠♦❞❡❧♦ ♦❢❡r❡❝❡ q✉❛♥❞♦ ❛♣❧✐❝❛❞♦ à s✐t✉❛çã♦ q✉❡ ♦ ♦r✐❣✐♥♦✉✱ ♥♦ s❡♥t✐❞♦ ❞❡ ✈❡r✐✜❝❛r ♦ q✉❛♥t♦ sã♦ ❛❞❡q✉❛❞❛s ♦✉ ♥ã♦✳
❚♦♠❛♥❞♦ ❝♦♠♦ ❜❛s❡ ❬●❆❩❩❊❚❚❆✱ ✶✾✽✾❪ ❡ ❬❇■❊▼❇❊◆●❯❚✱ ▼✳ ❙✳ ❍❊■◆✱ ✷✵✵✷❪ ♣♦❞❡♠♦s tr❛③❡r ✉♠❛ ❢♦r♠❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡♥s✐♥♦ ✉t✐❧✐③❛♥❞♦ ❛ ♠♦❞❡✲ ❧❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ❡♥❣❧♦❜❛♥❞♦ ❛❧❣✉♠❛s ❡t❛♣❛s q✉❡ sã♦✿
• ❈♦♠♣r❡❡♥sã♦ ❞❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛✿ ❖♥❞❡ ♦s ❛❧✉♥♦s ❡ ♦ ♣r♦❢❡ss♦r ❡s❝♦❧❤❡♠ ♦
t❡♠❛ ❡ ♦ ❢❡♥ô♠❡♥♦ ❛ s❡r tr❛❞✉③✐❞♦ ♣❛r❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳
• ❊str✉t✉r❛çã♦ ❞❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛✿ ❖♥❞❡ ♦s ✐♥❞✐✈í❞✉♦s ❡st✉❞❛♠ ♦ ❢❡♥ô♠❡♥♦ ❡
♣r♦♣õ❡ ❛s ❤✐♣ót❡s❡s ❡ ❢♦r♠✉❧❛♠ ♦ ♣r♦❜❧❡♠❛✳
• ▼❛t❡♠❛t✐③❛çã♦✿ ➱ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡st❡ ❡♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳
• ❙í♥t❡s❡✿ ❆♣ós ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♦❜❡❞❡❝❡♥❞♦ ❛s ❤✐♣ót❡s❡s ♣r♦♣♦st❛s✱ s✐♥✲
✷✸
• ■♥t❡r♣r❡t❛çã♦ ❡ ✈❛❧✐❞❛çã♦✿ ❆♣ós ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♦❜s❡r✈❛♠♦s s❡ ❡st❡s
s❡ ❛❞❡q✉❛♠ ❛♦ ❢❡♥ô♠❡♥♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛❞♦✳
• ❈♦♠✉♥✐❝❛çã♦ ❡ ❛r❣✉♠❡♥t❛çã♦✿ ◗✉❛♥❞♦ ❤á ✉♠❛ ❞✐s❝✉ssã♦ s♦❜r❡ ♦s r❡s✉❧t❛❞♦s
♦♥❞❡ ♦ ❣r✉♣♦ ✜❝❛ ❛ ♣❛r ❞♦s r❡s✉❧t❛❞♦s sã♦ ❝♦❧♦❝❛❞♦s ❛r❣✉♠❡♥t♦s s♦❜r❡ ❡st❡s ❡ ✈❡r✐✜❝❛✲s❡ s✉❛ ✈❛❧✐❞❛❞❡ ♣❛r❛ ♦ ❢❡♥ô♠❡♥♦ ♦❜s❡r✈❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡✳
❋✐❣✉r❛ ✶✳✶✿ ❊t❛♣❛s ❞❛ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ❡ ❛s ❆çõ❡s ❈♦❣♥✐t✐✈❛s ❞♦s ❆❧✉♥♦s ❬❉❊ ❆▲▼❊■❉❆✱ ▲✳ ▼ ❲✳❀ ❉❆ ❙■▲❱❆✱ ❑✳ P✳ ❈✐ê♥❝✐❛ &❊❞✉❝❛çã♦✱ ✈✳ ✶✽✱ ♥✳ ✸✱ ♣✳ ✻✸✷❪
❈♦♠ ♦ ♣❛ss❛r ❞♦s ❛♥♦s ♥♦t❛✲s❡ ✉♠ ❡s❢♦rç♦ ♣❛r❛ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ s❡❥❛ ❝♦♥str✉í❞♦ ❞✉r❛♥t❡ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♥♦ ❇r❛s✐❧ ❡ ♣❛r❛ q✉❡ ✐ss♦ ❛❝♦♥t❡ç❛ t❡♠♦s ❛❧✲ ❣✉♥s ❞♦❝✉♠❡♥t♦s ♥♦rt❡❛❞♦r❡s ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡♥tr❡ ❡❧❡s ♦s ♣r✐♥❝✐♣❛✐s sã♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭P❈◆✮ ✭❇❘❆❙■▲✱ ✷✵✵✵✮ ♥♦ q✉❛❧ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♦r✐❡♥t❛çõ❡s q✉❡ ♥ã♦ ♦❜❥❡t✐✈❛♠ ❛♣❡♥❛s ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❡ ❝♦♥t❡ú❞♦s✱ ♠❛s ❝♦♥s✐❞❡r❛♠ ✐♠♣♦rt❛♥t❡s t❛♠❜é♠ ❛s ❢♦r♠❛s ❞❡ ❡♥s✐♥❛r ❡ss❡s ❝♦♥t❡ú❞♦s✱ ❞❡ r❡✈❡❧❛r s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s ♥❡❝❡ssár✐❛s ❛ t♦❞♦ ❝✐❞❛❞ã♦✳
✷✹ ■♥tr♦❞✉çã♦
❯♠❛ ❡str❛té❣✐❛ ♣❛r❛ q✉❡ ❡ss❛s ❡str✉t✉r❛s s❡❥❛♠ ❝♦♥t❡♠♣❧❛❞❛s ❡ ♣❡❧♦ ❡♠♣r❡❣♦ ❞❡ ♠♦✲ ❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ q✉❡ ♣r❡s❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❡ s✉❣❡r❡ ❢♦r♠❛s ❞❡ ❡st✉❞♦ ❡ ❡♥s✐♥♦ ♣❛r❛ s♦❧✉❝✐♦♥á✲❧❛s✳ ❖s P❈◆s ❛✜r♠❛♠ q✉❡✿
❙❡ ♣♦r ✉♠ ❧❛❞♦ ❛ ✐❞❡✐❛ ❞❡ s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ ♣♦❞❡ ♣❛r❡❝❡r ♣❛r❛❞♦①❛❧✱ ♣♦✐s ❝♦♠♦ ♦ ❛❧✉♥♦ ♣♦❞❡ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ s❡ ❡❧❡ ♥ã♦ ❛♣r❡♥❞❡✉ ♦ ❝♦♥t❡ú❞♦ ♥❡✲ ❝❡ssár✐♦ à s✉❛ r❡s♦❧✉çã♦❄❀ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❤✐stór✐❛ ❞❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ♠♦str❛✲♥♦s q✉❡ ❡ss❡ ♠❡s♠♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❢♦✐ ❝♦♥str✉í❞♦ ❛ ♣❛rt✐r ❞❡ ♣r♦❜❧❡♠❛s ❛ s❡r❡♠ r❡s♦❧✈✐❞♦s✳ ✭❇❘❆❙■▲✱ ✷✵✶✷✱ ♣✳✽✹✮✳
▼❛s ♥ã♦ é só ♥♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ q✉❡ ♦ ✉s♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦t✐❞✐❛♥♦ t❡♠ s✐❞♦ r❡q✉✐s✐t❛❞♦✳ ■ss♦ t❛♠❜é♠ é ✈✐st♦ ♥♦ P❈◆ ❡♠ r❡❧❛çã♦ ❛ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s ❡s❝♦❧❛r❡s✱ ❝♦♠♦ ❛ ❢ís✐❝❛✱ ❛ q✉í♠✐❝❛✱ ❛ ❜✐♦❧♦❣✐❛✱ ❛ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛✳ ❊♠ ❝♦♥❥✉♥çã♦ ❝♦♠ ❡ss❡s ❡❧❡♠❡♥t♦s ❞♦ ❝♦t✐❞✐❛♥♦ ✈ê♠ ♦ ♣♦t❡♥❝✐❛❧ ♣❛r❛ ❛t✐✈✐❞❛❞❡s ✐♥t❡r❞✐s❝✐♣❧✐♥❛r❡s q✉❡ ❝❛❞❛ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r ❡st❛s ❞✐s❝✐♣❧✐♥❛s✱ ♣♦❞❡ ❛♣r❡s❡♥t❛r✳
❉✐❛♥t❡ ❞✐ss♦ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ❡str❛té❣✐❛ ❞❡ ❡♥s✐♥♦ s❡ r❡✈❡❧❛ ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ ❡ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ✉♠ ❡st✉❞♦ s♦❜r❡ ❡❧❛✱ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ ❞♦ q✉❡ ❛q✉❡❧❡ q✉❡ ✜③❡♠♦s ❡♥q✉❛♥t♦ ❧✐❝❡♥❝✐❛♥❞♦ ❡♠ ♠❛t❡♠át✐❝❛✱ s❡ ♠♦str❛ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ♣r♦✜s✲ s✐♦♥❛❧ ❞❛ ❡❞✉❝❛çã♦ q✉❡ ♣r❡t❡♥❞❡ ❛t❡♥❞❡r ❛s ❞❡♠❛♥❞❛s ❛t✉❛✐s ❞♦ ❊♥s✐♥♦✳
❘❡❝♦♥❤❡❝❡r ❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❡ ❛ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ❡str❛té❣✐❛ ❞❡ tr❛❞✉çã♦✱ é ✉♠ ♣❛r❛♠❡tr♦ q✉❡ ♦ P❈◆ ❛♣r❡s❡♥t❛✱ ❜❡♥❡✜❝✐❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♣♦♣✉❧❛çã♦✱ q✉❡ ❡st❛ ❡♠ ❝♦♥t✐♥✉♦ ♣r♦❝❡ss♦ ❞❡ ❡✈♦❧✉çã♦ ❡ ♥❡❝❡ss✐t❛ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡ s❡ t♦r♥❛r ✢✉❡♥t❡ ♥❡st❛ tã♦ ✐♠♣♦rt❛♥t❡ ❧✐♥❣✉❛❣❡♠✳
❆❝r❡❞✐t❛♠♦s q✉❡ ✉♠❛ ❛✉❧❛ ❡♠ q✉❡ s❡ ❡♠♣r❡❣❛ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ❡str❛té❣✐❛ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ♣♦❞❡ s❡r ❡✜❝❛③ ❡ ❛ ❧✐t❡r❛t✉r❛ ❝♦♠♣r♦✈❛ ✐ss♦ ♣♦r ♠❡✐♦ ❞❡ r❡❧❛t♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ❡ ❛ ♣❡sq✉✐s❛ ❞❡ ♥♦✈❛s ❡str❛té❣✐❛s ♣❛r❛ tr❛❜❛❧❤á✲❧❛ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✳
✷ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❛ r❡s♣❡✐t♦ ❞❡ s❡q✉ê♥❝✐❛s ♥✉✲ ♠ér✐❝❛s✳ ❉✐s❝✉t✐r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞♦s ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ❡ ♦s ❝r✐tér✐♦s ❞❡ ❝♦♥✈❡r❣❡♥❝✐❛✳ P❛r❛ ✐ss♦✱ s❡❣✉✐r❡♠♦s ❛ r❡❢❡rê♥❝✐❛ ❬▼❆❚❖❙✱ ▼✳P✳❪✳
✷✳✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s
❘❡♣r❡s❡♥t❛♠♦s ♣♦rN ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ✐st♦ é✱ N={1,2,3,4, . . .}.
❊ss❡ ❝♦♥❥✉♥t♦ é ❞❡♥♦♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❖ s✉❜❝♦♥❥✉♥t♦ ❞❡N′
❝♦♥st✐t✉í❞♦ ❞♦s ♥ú♠❡r♦s ♣❛r❡s s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r Np ❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s í♠♣❛r❡s✱ ♣♦r NI✳ ❊♠ sí♠❜♦❧♦s✱ ❡s❝r❡✈❡♠♦s✿
Np ={2n;n ∈N} ❡ NI ={2n−1;n ∈N}.
❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ ♦✉ s✉❝❡ssã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❢✉♥çã♦ f :N→R✱
q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ✉♠ ♥ú♠❡r♦ r❡❛❧ f(n)✳
❖ ✈❛❧♦r ❞❛ s❡q✉ê♥❝✐❛f ♥♦ ♥ú♠❡r♦ ♥❛t✉r❛❧né ❞❡♥♦♠✐♥❛❞♦n✲és✐♠♦ t❡r♠♦ ♦✉ t❡r♠♦
❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛f❡ é r❡♣r❡s❡♥t❛❞♦ ❣❡♥❡r✐❝❛♠❡♥t❡ ♣♦ran✱bn✱xn❡t❝✳ P❛r❛ s✐♠♣❧✐✜❝❛r✱ ❢❛r❡♠♦s r❡❢❡rê♥❝✐❛ ❛♦ t❡r♠♦ ❣❡r❛❧ an ❝♦♠♦ ❛ s❡q✉ê♥❝✐❛ f t❛❧ q✉❡ f(n) = an✳ ❯♠❛ s❡q✉ê♥❝✐❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ s❡✉ t❡r♠♦ ❣❡r❛❧ ♦✉ ❡①♣❧✐❝✐t❛♥❞♦✲s❡ s❡✉s ♣r✐♠❡✐r♦s t❡r♠♦s✳ ❆ s❡❣✉✐r t❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s❡q✉ê♥❝✐❛s✿
❊①❡♠♣❧♦ ✷✳✶✳ an = 1 + (−1)n+1 = (2,0,2,0,· · ·)✳ ❊①❡♠♣❧♦ ✷✳✷✳ an =n = (1,2,3,· · ·)✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ f : N → R✱ ❛s r❡str✐çõ❡s ❞❡ f ❛ s✉❜❝♦♥❥✉♥t♦s
✐♥✜♥✐t♦s ❞❡ N s❡rã♦ ❞❡♥♦♠✐♥❛❞❛s s✉❜s❡q✉ê♥❝✐❛s ❞❡ f✳
❘❡♣r❡s❡♥t❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ f ♣❡❧♦ s❡✉ t❡r♠♦ ❣❡r❛❧ {an}✱ n ∈ N✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛s s✉❜s❡q✉ê♥❝✐❛s ❞❡f ♦✉ ❞❡{an}✱ sã♦ ❛q✉❡❧❛s s❡q✉ê♥❝✐❛s{ak}✱ ❝♦♠k ∈N
′ ♦♥❞❡ ◆✬
é s✉❜❝♦♥❥✉♥t♦ ❞❡ N✳ ◆❛t✉r❛❧♠❡♥t❡✱ t♦❞❛ s❡q✉ê♥❝✐❛ é ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡❧❛ ♣ró♣r✐❛✳
❉❡♥tr❡ ❛s s✉❜s❡q✉ê♥❝✐❛s ❞❡ ✉♠❛ ❞❛❞❛ s❡q✉ê♥❝✐❛{an}❞❡st❛❝❛♠♦s ❞✉❛s✿ ❛ s✉❜s❡q✉ê♥❝✐❛ ♣❛r {a2n} ❡ ❛ s✉❜s❡q✉ê♥❝✐❛ í♠♣❛r {a2n−1}✳
✷✻ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
❊①❡♠♣❧♦ ✷✳✸✳ ❆s s✉❜s❡q✉ê♥❝✐❛s ♣❛r ❡ í♠♣❛r ❞❛ s❡q✉ê♥❝✐❛an= (−1)nsã♦ ❛s s❡q✉ê♥✲ ❝✐❛s ❝♦♥st❛♥t❡s a2n = 1 ❡ a2n−1 =−1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✷✳✷ ▲✐♠✐t❛çã♦ ❡ ▼♦♥♦t♦♥✐❛
❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠❛ s❡q✉ê♥❝✐❛ {an} é ❞✐t❛ ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ M✱ ❞❡♥♦♠✐♥❛❞♦ ❝♦t❛ s✉♣❡r✐♦r ❞❛ s❡q✉ê♥❝✐❛✱ q✉❡ ❛t❡♥❞❡ à s❡❣✉✐♥t❡ ❝♦♥❞✐✲
çã♦✿
an≤M, ♣❛r❛ t♦❞♦ n ∈N.
❉❡✜♥✐çã♦ ✷✳✹✳ ❯♠❛ s❡q✉ê♥❝✐❛ {an} é ❞✐t❛ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧m✱ ❞❡♥♦♠✐♥❛❞♦ ❝♦t❛ ✐♥❢❡r✐♦r ❞❛ s❡q✉ê♥❝✐❛✱ q✉❡ ❛t❡♥❞❡ à s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
m≤an, ♣❛r❛ t♦❞♦ n∈N.
❉❡✜♥✐çã♦ ✷✳✺✳ ❯♠❛ s❡q✉ê♥❝✐❛ {an} é ❞✐t❛ ❧✐♠✐t❛❞❛ q✉❛♥❞♦ ♦ ❢♦r s✉♣❡r✐♦r♠❡♥t❡ ❡ ✐♥❢❡r✐♦r♠❡♥t❡✱ ✐st♦ é✱ q✉❛♥❞♦ ❡①✐st✐r ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C t❛❧ q✉❡✿
|an| ≤C, ♣❛r❛ t♦❞♦ n∈N.
➱ ❝❧❛r♦ q✉❡ s❡ M ❢♦r ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ ✉♠❛ ❞❛❞❛ s❡q✉ê♥❝✐❛ {an}✱ ❡♥tã♦ q✉❛❧✲ q✉❡r ♥ú♠❡r♦ r❡❛❧ ♠❛✐♦r ❞♦ q✉❡ M t❛♠❜é♠ s❡rá ❝♦t❛ s✉♣❡r✐♦r ❞❛ s❡q✉ê♥❝✐❛ {an}✳ ❆ ♠❡♥♦r ❞❡ss❛s ❝♦t❛s s✉♣❡r✐♦r❡s é ❞❡♥♦♠✐♥❛❞❛ s✉♣r❡♠♦ ❞❛ s❡q✉ê♥❝✐❛ {an} ❡ ❞❡♥♦t❛❞❛ ♣♦r sup{an}✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✉♠❛ s❡q✉ê♥❝✐❛ {an} ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦ss✉✐ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❝♦t❛s ✐♥❢❡r✐♦r❡s✱ s❡♥❞♦ ❛ ♠❛✐♦r ❞❡❧❛s ❞❡♥♦♠✐♥❛❞❛s ❞❡ í♥✜♠♦ ❞❛ s❡q✉ê♥❝✐❛ ❡ ❞❡♥♦t❛❞❛ ♣♦rinf{an}✳ ◆♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ ♣r❡❝✐s❛♠♦s ❞❡✐①❛r ❝❧❛r♦ ❞♦✐s ❢❛t♦s ❢✉♥✲ ❞❛♠❡♥t❛✐s✿ ♣r✐♠❡✐r♦✱ q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ t❡♠ s✉♣r❡♠♦ ✜♥✐t♦ ❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ t❡♠ í♥✜♠♦ ✜♥✐t♦❀ ❞❡♣♦✐s✱ q✉❡ ♣❛r❛ ❝❛❞❛ ǫ > 0
♦ ♥ú♠❡r♦ r❡❛❧ α = sup{an} −ǫ✱ ♣♦r s❡r ♠❡♥♦r ❞♦ q✉❡ ♦ s✉♣r❡♠♦ ❞❛ s❡q✉ê♥❝✐❛ {an}✱ ♥ã♦ ♣♦❞❡ s❡r ❛ ❝♦t❛ s✉♣❡r✐♦r ❡✱ ♣♦r ❡ss❛ r❛③ã♦✱ ❡①✐st❡ ❛❧❣✉♠ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♣♦r ❡①❡♠♣❧♦ an1✱ t❛❧ q✉❡✿
α= sup{an} −ǫ < an1 .
P❛r❛ ♦ í♥✜♠♦ ♦❝♦rr❡ ✉♠ ❢❛t♦ ❛♥á❧♦❣♦✳ ❙❡♥❞♦β = inf {an}+ǫ✉♠ ♥ú♠❡r♦ r❡❛❧ ♠❛✐♦r ❞♦ q✉❡ ♦ í♥✜♠♦ ❞❛ s❡q✉ê♥❝✐❛{an}✱ ❡①✐st❡ ❛❧❣✉♠ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♣♦r ❡①❡♠♣❧♦an2✱ é t❛❧ q✉❡✿
β = inf {an}+ǫ > an2 .
❊①❡♠♣❧♦ ✷✳✹✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦ ❣❡r❛❧ an =n é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♠❛s ♥ã♦ s✉♣❡r✐♦r♠❡♥t❡✳ ❚❡♠♦s q✉❡ inf{an}= 1✳
❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s ✷✼
❊①❡♠♣❧♦ ✷✳✻✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦ ❣❡r❛❧ an= (−1)n é ❧✐♠✐t❛❞❛✱ s❡♥❞♦sup{an}= 1 ❡ inf{an}=−1✳
❉❡✜♥✐çã♦ ✷✳✻✳ ❯♠❛ s❡q✉ê♥❝✐❛ {an} é ❞❡♥♦♠✐♥❛❞❛ ♠♦♥ót♦♥❛ ❝r❡s❝❡♥t❡ ♦✉ ♥ã♦ ❞❡✲ ❝r❡s❝❡♥t❡ q✉❛♥❞♦ an≤an+1✱ ♣❛r❛ t♦❞♦ n∈N✱ ✐st♦ é✱ a1 ≤a2 ≤a3 ≤. . .✳
❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠❛ s❡q✉ê♥❝✐❛{an}é ❞❡♥♦♠✐♥❛❞❛ ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ♦✉ ♥ã♦ ❝r❡s✲ ❝❡♥t❡s q✉❛♥❞♦ an+1 ≤an✱ ♣❛r❛ t♦❞♦ n ∈N✱ ✐♥t♦ é a1 ≥a2 ≥a3 ≥. . .✳
❊①❡♠♣❧♦ ✷✳✼✳ ❆s s❡q✉ê♥❝✐❛s an=n ❡ bn = lnn sã♦ ❝r❡s❝❡♥t❡s✱ ❡♥q✉❛♥t♦ cn =−n3 ❡
dn = n1 sã♦ ❞❡❝r❡s❝❡♥t❡s✳
❊①❡♠♣❧♦ ✷✳✽✳ ❆ s❡q✉ê♥❝✐❛ an =
n
n+ 1 é ❝r❡s❝❡♥t❡✳ ❉❡ ❢❛t♦✱ an+1
an
= n+ 1 n+ 2 ·
n+ 1 n =
n2
+ 2n+ 1
n2+ 2n ≥1, ♣❛r❛ t♦❞♦n ∈N,
❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ an+1 ≥an ✱ ♣❛r❛ t♦❞♦ n∈N✳
✷✳✸ ❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s
❉❡✜♥✐çã♦ ✷✳✽✳ ❉✐③❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ l é ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ {an}✱ ♦✉ q✉❡ ❛ s❡q✉ê♥❝✐❛ {an} ❝♦♥✈❡r❣❡ ♣❛r❛ l✱ q✉❛♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❢♦r ❛t❡♥❞✐❞❛✿
♣❛r❛ t♦❞♦ ǫ >0, ❡①✐st❡ n0 ∈N t❛❧ q✉❡ |an−l|< ǫ, ♣❛r❛ t♦❞♦ n≥n0.
❆q✉✐✱ ❞❡♥♦t❛♠♦s
lim
n→∞an=l.
❛✮ ❖ ♥ú♠❡r♦ ♥❛t✉r❛❧n0 ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ❡♠ ❣❡r❛❧ ❞❡♣❡♥❞❡ ❞♦ ♥ú♠❡r♦ ǫ❞❛❞♦❀
❜✮ ❙❡♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ |an−l|< ǫ ❡q✉✐✈❛❧❡♥t❡ ❛
l−ǫ < an< l+ǫ, ✭✷✳✶✮ ♦✉ ❛✐♥❞❛ q✉❡ an ∈ (l−ǫ, l+ǫ) ❛ s❡♥t❡♥ç❛ ✭✷✳✶✮ ❡st❛❜❡❧❡❝❡ q✉❡ ❢♦r❛ ❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦(l−ǫ, l+ǫ)❡①✐st❡ ♥♦ ♠á①✐♠♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛
♦✉✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ q✉❡ t♦❞♦s ♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❛ ♣❛rt✐r ❞♦ t❡r♠♦ ❞❡ ♦r❞❡♠ n0 ❡stã♦ ❞❡♥tr♦ ❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (l−ǫ, l+ǫ)❀
❝✮ ❆ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦ sã♦ ❛❧t❡r❛❞♦s q✉❛♥❞♦ s❡ ✏♠❡①❡✑ ❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ t❡r♠♦s✳ ❆q✉✐✱ ✏♠❡①❡r✑ s✐❣♥✐✜❝❛ ❛❝r❡s❝❡♥t❛r✱ ♦♠✐t✐r ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♠✉❞❛r ♦ ✈❛❧♦r✳ P♦r ❡ss❛ r❛③ã♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s✉❛ ❝❛✉❞❛✶❀
✷✽ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
❞✮ ❋✐♥❛❧♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ♥ã♦ ♣♦❞❡ t❡r ❞♦✐s ❧✐♠✐t❡s✳ ❉❡ ❢❛t♦✱ s❡
l1 = lim
n→∞an ❡ l2 = limn→∞an,
❡♥tã♦✱ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ǫ✱ ❡①✐st❡♠✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ✷✳✽✱
❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐sn1 ❡n2 t❛✐s q✉❡
|an−l1|<
ǫ
2, ♣❛r❛ t♦❞♦n≥n1 ❡ |an−l2|< ǫ
2, ♣❛r❛ t♦❞♦n≥n2.
P❛r❛ q✉❡ ❡ss❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♦❝♦rr❛♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ é s✉✜❝✐❡♥t❡ ❝♦♥s✐❞❡✲ r❛r♠♦s ✉♠ í♥❞✐❝❡ q✉❡ s❡❥❛ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ n1 ❡ n2 ❛♦ ♠❡s♠♦ t❡♠♣♦✳ ❙❡
n3 = max{n1, n2} é ✉♠ t❛❧ í♥❞✐❝❡✱ t❡♠♦s✿
|l1−l2| ≤ |an3 −l1|+|an3 −l2|< ǫ ✭✷✳✷✮
❡ ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ só é ♣♦ssí✈❡❧ ♣❛r❛ q✉❛❧q✉❡r ǫ ♣♦s✐t✐✈♦ q✉❛♥❞♦ l1 =l2✱ r❡s✉❧✲
t❛♥❞♦ ♥❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✳
❊①❡♠♣❧♦ ✷✳✾✳ ❈♦♠♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦✱ ✈❡r✐✜❝❛r❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ 1
n ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✱ ✐st♦ é
lim
n→∞
1 n = 0.
◆❡ss❡ ❝❛s♦✱l = 0✱ an =
1
n ❡✱ ♣❛r❛ ✐❧✉str❛r ❛ ❞❡✜♥✐çã♦ ✷✳✽✱ ❝♦♥s✐❞❡r❡♠♦s ǫ= 0.01✳ P❛r❛
♦❝♦rr❡r
1 n −0
<0.01,
❜❛st❛ ❝♦♥s✐❞❡r❛r♠♦sn >100✱ ❥á q✉❡ ✐st♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ 1
n <0.01✳ ◆❡ss❡ ❝❛s♦n0 = 101
s❛t✐s❢❛③ (2.1)✳ ◆♦ ❝❛s♦ ❣❡r❛❧✱ ❞❛❞♦ ǫ > 0✱ ❡s❝♦❧❤❛♠♦s n0 ∈ N t❛❧ q✉❡ n0 >
1
ǫ✳ ➱ ❝❧❛r♦
q✉❡✱ s❡ n ≥ n0✱ ❡♥tã♦
1 n ≤
1 n0
< ǫ ❡ ✐st♦ ❛❝❛rr❡t❛
1 n −0
< ǫ✳ ❊st❡ é ✉♠ ❡①❡♠♣❧♦ ❞❡
✉♠❛ s❡q✉ê♥❝✐❛ ✭❞❡❝r❡s❝❡♥t❡✮ ❝♦♥✈❡r❣❡♥t❡✳
❊①❡♠♣❧♦ ✷✳✶✵✳ ❆ s❡q✉ê♥❝✐❛ ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é an =
2n2
n2
−4✱ ❝♦♠ n ≥ 3✱ ❝♦♥✈❡r❣❡
♣❛r❛ 2✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ ǫ >0✱ t❡♠♦s✿
2n2
n2−4−2
< ǫ⇔
2n2
−2n2
+ 8 n2 −4
< ǫ⇔
8 n2−4
< ǫ. ✭✷✳✸✮
P❛r❛ n≥3✱ t❡♠♦s n2
−4≥5 ❡ ♣❛r❛ ❡st❛❜❡❧❡❝❡r (2.3)❜❛st❛ ❝♦♥s✐❞❡r❛r♠♦s
n2
−4> 8
ǫ ♦✉ n >
r
4 + 8 ǫ.
■ss♦ ♥♦s s✉❣❡r❡ ❡s❝♦❧❤❡r n0 ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ t❛❧ q✉❡ n0 >
q
4 + 8
❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s ✷✾
❱❡❥❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞❡ s❡q✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✷✳✶✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞❛✳
❙❡❥❛{an} ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❝♦♠ ❧✐♠t❡l✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❛ ǫ= 1✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡n0 ❛ ♣❛rt✐r ❞♦ q✉❛❧ s❡ t❡♠|an−l|<1✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣♦❞❡♠♦s ❛ss❡❣✉r❛r q✉❡✿
|an|=|an−l+l| ≤ |an−l|+|l|<1 +l, ♣❛r❛ t♦❞♦n ≥n0.
❖s ú♥✐❝♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ q✉❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ♥ã♦ ❛t❡♥❞❡♠ à ❝♦♥❞✐çã♦ ❛❝✐♠❛ sã♦✿ a1, a2, a3, . . . , an0−1✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♥ú♠❡r♦ r❡❛❧ C ❝♦♠♦ ♦ ♠❛✐♦r ❡♥tr❡ ♦s
♥ú♠❡r♦s 1 +l, |a1|, |a2|, |a3|, . . . , |an0−1|✱ t❡r❡♠♦s✿
|an| ≤C, ♣❛r❛ t♦❞♦n.
❚❡♦r❡♠❛ ✷✳✷✳ ✭❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss✮ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳
❙❡❥❛ {an} ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✿
M ={n∈N;an> am, ∀m > n}.
❚❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛M✿ M é ✐♥✜♥✐t♦ ♦✉M é ✜♥✐t♦✳
✐✮ M é ✐♥✜♥✐t♦✿ ❊s❝r❡✈❛♠♦s M ={n1, n2, n3,· · · } ❝♦♠ n1 < n2 < n3 <· · · ✳❆ss✐♠✱
s❡ i < j ❡♥tã♦ ni < nj ❡✱ ❝♦♠♦ ni ∈ M✱ ♦❜t❡♠♦s q✉❡ ani > anj ✳ ❈♦♥❝❧✉í♠♦s ❡♥tã♦✱ q✉❡ ❛ s✉❜s❡q✉ê♥❝✐❛ {ank} é ❞❡❝r❡s❝❡♥t❡✳ ❙❡♥❞♦ ❡❧❛ ❧✐♠✐t❛❞❛ ♦❜t❡♠♦s✱ ✜♥❛❧♠❡♥t❡✱ q✉❡ ❡❧❛ é ❝♦♥✈❡r❣❡♥t❡✳
✐✐✮ M é ✜♥✐t♦✿ ❈♦♠♦ M é ✜♥✐t♦✱ ❡①✐st❡ n1 ∈ N ⊂ M ❝♦t❛ s✉♣❡r✐♦r ❞❡ M✳ ❖r❛✱
n1 ∈ M ❧♦❣♦✱ ❡①✐st❡ n2 > n1✭❡ ♣♦rt❛♥t♦ n2 ∈ M✮ t❛❧ q✉❡ an1 ≤ an2 ✳ ▼❛s ❞❡ n2 ∈ M s❡❣✉❡ q✉❡ ❡①✐st❡ n3 > n2 ✭❡ ♣♦rt❛♥t♦ n3 ∈ M✮ t❛❧ q✉❡ an2 ≤ an3 ✳ P♦r ✐♥❞✉çã♦✱ ❞❡✜♥✐♠♦s ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ {ank} q✉❡ é ❝r❡s❝❡♥t❡ ❡✱ ♣♦rt❛♥t♦✱ ❝♦♥✈❡r❣❡♥t❡ ✭♣♦✐s ❡❧❛ é ❧✐♠✐t❛❞❛✮✳
❚❡♦r❡♠❛ ✷✳✸✳ ❙❡ f : [a,∞) → R é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡ limx→∞f(x) = l✱ ❡♥tã♦ ❛
s❡q✉ê♥❝✐❛ an =f(n)✱ n > a✱ é ❝♦♥✈❡r❣❡♥t❡ ❡ s❡✉ ❧✐♠✐t❡ é ✐❣✉❛❧ ❛l✳ ❙❡ lim
x→∞f(x) =±∞✱
❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ {an} é ❞✐✈❡r❣❡♥t❡✳
✸✵ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
x ≥ K✳ P❛ss❛♥❞♦ ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ❞❡ s❡q✉ê♥❝✐❛s✱ ❡s❝♦❧❤❡♠♦s ✉♠ í♥❞✐❝❡ n0 ≥ K ❡
t❡r❡♠♦s|f(n)−l|< ǫ✱ ♣❛r❛ t♦❞♦ n≥n0✳
◆♦s ❝❛s♦s ❡♠ q✉❡ é ♣♦ssí✈❡❧ ✉t✐❧✐③❛r ♦ ❚❡♦r❡♠❛ ✷✳✸✱ ♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s t♦r♥❛✲s❡ r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ q✉❛♥❞♦ s❡ ✉s❛♠ ❚é❝♥✐❝❛s ❞❡ ❈á❧❝✉❧♦✱ ❛ ❡①❡♠♣❧♦ ❞❛ ❢❛♠♦s❛ r❡❣r❛ ❞❡ ▲✬❍ô♣✐t❛❧✳
❊①❡♠♣❧♦ ✷✳✶✶✳ P❛r❛ ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛an =
lnn
n ✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❢✉♥çã♦
❡①t❡♥sã♦ f(x) = lnx
x ✱ ❞❡✜♥✐çã♦ ♣❛r❛ x > 0✱ ❡ ✐♥✈❡st✐❣❛♠♦s s❡✉ ❧✐♠✐t❡ ♥♦ ✐♥✜♥✐t♦✳ ❉❛
❘❡❣r❛ ❞❡ ▲✬❍ô♣✐t❛❧ r❡s✉❧t❛ q✉❡✿
lim
x→∞
lnx
x = limx→∞
1 x = 0
❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✸ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ lnn
n ❝♦♥✈❡r❣❡ ♣❛r❛ 0✳
❱❡❥❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ♣❛r❛ ♦ ❧✐♠✐t❡ ❞❡ s❡q✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✷✳✹✳ ✭Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐♠✐t❡✮ ❙❡❥❛♠ {an} ❡ {bn} s❡q✉ê♥❝✐❛s ❝♦♥✈❡r❣❡♥t❡s ❝♦♠ ❧✐♠✐t❡ l ❡ r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦
✭❛✮ ❛ s❡q✉ê♥❝✐❛ {an±bn} ❝♦♥✈❡r❣❡ ♣❛r❛ l±r❀
✭❜✮ ❛ s❡q✉ê♥❝✐❛ {c·an} ❝♦♥✈❡r❣❡ ♣❛r❛ c·l✱ ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡❀ ✭❝✮ ❛ s❡q✉ê♥❝✐❛ {|an|} ❝♦♥✈❡r❣❡ ♣❛r❛ |l|❀
✭❞✮ ❛ s❡q✉ê♥❝✐❛ {an·bn} ❝♦♥✈❡r❣❡ ♣❛r❛ l·r❀
✭❡✮ ❛ s❡q✉ê♥❝✐❛ {an/bn} ❝♦♥✈❡r❣❡ ♣❛r❛ l/r✱ q✉❛♥❞♦ r6= 0 ❡ bn6= 0✱ ∀n❀
❙❡❥❛ǫ >0 ❞❛❞♦✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡✱ ❡①✐st❡♠ í♥❞✐❝❡sn1 ❡ n2 t❛✐s q✉❡✿
|an−l|< ǫ,♣❛r❛ t♦❞♦n≥n1 ✭✷✳✹✮
|an−r|< ǫ, ♣❛r❛ t♦❞♦n ≥n2 ✭✷✳✺✮
❡ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ í♥❞✐❝❡ n0 = max{n1, n2} ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ n1 ❡ n2✱ ❞❡ ♠♦❞♦
q✉❡ (2.4) ❡(2.5)♦❝♦rr❛♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ t❡♠♦s ♣❛r❛ n ≥n0 q✉❡✿
✭❛✮ |an±bn−(l±r)| ≤ |an−l|+|bn−r|< ǫ+ǫ= 2ǫ❀ ✭❜✮ |can−cl|=|c||an−l|<|c|ǫ❀
✭❝✮ ||an| − |l|| ≤ |an−l|< ǫ❀
✭❞✮ |anbn−lr|=|anbn−bnl+bnl−lr| ≤ |bn||an−l|+|l||bn−r| ≤(C+|l|)ǫ✱ ♦♥❞❡ C é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ q✉❡ ❧✐♠✐t❛ ❛ s❡q✉ê♥❝✐❛{bn}✳ ❆ ❡①✐stê♥❝✐❛ ❞❛ ❝♦♥st❛♥t❡
❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s ✸✶
✭❡✮ ❊st❡ ❝❛s♦ é ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡❧✐❝❛❞♦✳ ❙❡❥❛C✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ 1
|bn| ≤
C✱
♣❛r❛ t♦❞♦ n≥n0✳ ❚❡♠♦s✳
an
bn −
l r =
anr−bnl−lr+lr
rbn ≤ 1
|bn|
|an−l|+ |
l|
|r||bn−r|
< C
1 + |l|
|r|
ǫ.
❉❡ ♣♦ss❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛♣r❡s❡♥t❛❞❛s ♥♦ ❚❡♦r❡♠❛ ✷✳✹✱ ✜❝❛ ♠❛✐s ♣rát✐❝♦ ♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s✳ ◆ã♦ é ♠❛✐s ♥❡❝❡ssár✐♦ ✐♥tr♦❞✉③✐r ❛ ❢✉♥çã♦ ❡①t❡♥sã♦ f(x)✱ ❛ ♠❡♥♦s q✉❡ s❡
❢❛ç❛ r❡❢❡rê♥❝✐❛ às s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❛♥❛❧ít✐❝❛s ❝♦♠♦ ❝♦♥t✐♥✉✐❞❛❞❡✱ ❞❡r✐✈❛❜✐❧✐❞❛❞❡ ✱ ❡t❝✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❝❛❧❝✉❧❛r
lim
n→∞
n2
+ 3 4n2
−2n+ 1,
❝♦❧♦❝❛♠♦s ❡♠ ❡✈✐❞ê♥❝✐❛ ♦ t❡r♠♦ ❞❡ ♠❛✐♦r ❣r❛✉✱ r❡s✉❧t❛♥❞♦✿
lim
n→∞
n2
(1 + 3
n2)
n2(4− 2
n+
1
n2)
= lim
n→∞
1 + 3
n2
4− 2
n +
1
n2
= 1
4.
❱❡❥❛♠♦s ♠❛✐s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ s❡q✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✷✳✺✳ ❙❡ ✉♠❛ s❡q✉ê♥❝✐❛ {an} ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ ❡ {bn} é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ ♣r♦❞✉t♦ {an·bn} ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✳
❙❡❥❛ǫ >0❞❛❞♦✳ ❈♦♠♦ ❛ s❡q✉ê♥❝✐❛{an}❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✱ ❛ ❡st❡ǫ❝♦rr❡s♣♦♥❞❡ ✉♠ í♥❞✐❝❡ n0 t❛❧ q✉❡ |an| < ǫ✱ s❡♠♣r❡ q✉❡ n ≥ n0✳ ❖r❛✱ s❡♥❞♦ {bn} ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C t❛❧ q✉❡ |bn| ≤C✱ s❡❥❛ q✉❛❧ ❢♦r ♦ í♥❞✐❝❡ n✱ ❡ ❝❡rt❛♠❡♥t❡ ♣❛r❛ q✉❛❧q✉❡r n≥n0 t❡r❡♠♦s✿
|anbn−0|=|anbn|=|an||bn|< Cǫ.
◆♦ ❚❡♦r❡♠❛ ✷✳✺✱ ❛ s❡q✉ê♥❝✐❛ {bn} é ❛♣❡♥❛s ❧✐♠✐t❛❞❛✱ ♣♦❞❡♥❞♦ s❡r ❝♦♥✈❡r❣❡♥t❡ ♦✉ ♥ã♦✳ P♦r ❡ss❛ r❛③ã♦ ♥ã♦ ❢♦✐ ✉s❛❞❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❧✐♠✐t❡ r❡❢❡r❡♥t❡ ❛♦ ♣r♦❞✉t♦ ❞❡ s❡q✉ê♥❝✐❛s✱ ❛ q✉❛❧ ❡①✐❣❡ ❛ ❡①✐stê♥❝✐❛ ❞♦s ❧✐♠✐t❡s ❞❛s s❡q✉ê♥❝✐❛s ❡♥✈♦❧✈✐❞❛s✳
❊①❡♠♣❧♦ ✷✳✶✷✳ ❙❡ an =
1
n ❡ bn = sen(n) ❡♥tã♦ {bn} ♥ã♦ ❝♦♥✈❡r❣❡ ♠❛s✱ ❝♦♠♦ −1 ≤
bn≤1✱ t❡♠✲s❡ lim
n→∞anbn = limn→∞
sen(n)
n = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡nlim→∞an = 0♠❛s bn ♥ã♦
é ❧✐♠✐t❛❞❛✱ ♦ ♣r♦❞✉t♦ anbn ♣♦❞❡ ❞✐✈❡r❣✐r ✭♣♦r ❡①❡♠♣❧♦ an =
1
n✱ bn =n
2✮ ♦✉ ❝♦♥✈❡r❣✐r
♣❛r❛ ✉♠ ✈❛❧♦r q✉❛❧q✉❡r ✭♣♦r ❡①❡♠♣❧♦ an= n1✱ bn =c.n)✳
✸✷ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
❚❡♦r❡♠❛ ✷✳✻✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ q✉❡ é ❛♦ ♠❡s♠♦ ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ❡ ❧✐♠✐t❛❞❛ ❝♦♥✲ ✈❡r❣❡ ♣❛r❛ ♦ s❡✉ í♥✜♠♦✳
❙❡❥❛ ❡♥tã♦{an}✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ ❡ s❡❥❛l= inf{an}✳ ❉❛❞♦ǫ >0✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡ n0 t❛❧ q✉❡ l+ǫ > an0 ❡✱ s❡♥❞♦ ❛ s❡q✉ê♥❝✐❛ {an} ❞❡❝r❡s❝❡♥t❡✱ t❡♠♦s✿
an ≤an0 < l+ǫ, ♣❛r❛ t♦❞♦n ≥n0.
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ l é ♦ í♥✜♠♦ ❞❛ s❡q✉ê♥❝✐❛ {an}✱ ❡♥tã♦ l≤an✱ ♣❛r❛ t♦❞♦n✱ ❞❡ ♠♦❞♦ q✉❡✿
l−ǫ < l ≤an, ♣❛r❛ t♦❞♦n. ❈♦♠❜✐♥❛♥❞♦ ❛s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛✱ ♦❜t❡♠♦s✿
l−ǫ < an< l+ǫ, ♣❛r❛ t♦❞♦n≥n0
❡ ✐st♦ é ❡q✉✐✈❛❧❡♥t❡ ❛✿
|an−l|< ǫ, ♣❛r❛ t♦❞♦n ≥n0.
❈♦♠ ✐ss♦ ♣r♦✈❛♠♦s q✉❡
lim
n→∞an=l.
❊①❡♠♣❧♦ ✷✳✶✸✳ ❆ s❡q✉ê♥❝✐❛ ❝✉❥♦ ♥✲és✐♠♦ t❡r♠♦ é an =
1
n é ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ❡ ❧✐♠✐t❛❞❛✳ ❊♥tã♦✱ t❡♠♦s
lim
n→∞
1 n = inf
1
n;n ∈N
= 0.
❚❡♦r❡♠❛ ✷✳✼✳ ✭❙❛♥❞✉í❝❤❡✮
❙❡❥❛♠ {an}✱ {bn} ❡ {cn} três s❡q✉ê♥❝✐❛s q✉❡ s❛t✐s❢❛③❡♠ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
an≤bn≤cn, ∀n. ❙❡
lim
n→∞an= limn→∞cn=l,
❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ {bn} é ❝♦♥✈❡r❣❡♥t❡ ❡ s❡✉ ❧✐♠✐t❡ é ✐❣✉❛❧ ❛ l✳ ❉❡♠♦s♥tr❛çã♦✿
❉❛❞♦ ǫ >0✱ s❡❥❛ n0 ∈N ✉♠ í♥❞✐❝❡ ❛ ♣❛rt✐r ❞♦ q✉❛❧ s❡ t❡♠✿
−ǫ < an−l < ǫ ❡ −ǫ < cn−l < ǫ.
◆♦t❛♥❞♦✱ q✉❡ an−l≤bn−l ≤cn−l ❡ ✉s❛♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛✱ ♦❜t❡♠♦s✿
−ǫ < bn−l < ǫ, ♣❛r❛ t♦❞♦n≥n0,
♦✉✱ ❞❡ ♠♦❞♦ ❡q✉✐✈❛❧❡♥t❡✱ |bn−l|< ǫ✱ ♣❛r❛ t♦❞♦ n ≥n0✳
❙❡q✉ê♥❝✐❛s ❈♦♥✈❡r❣❡♥t❡s ✸✸
❊①❡♠♣❧♦ ✷✳✶✹✳ ❙❡❥❛ an =
cosn
n ✳ ❈♦♠♦ −1 ≤ cosn ≤ 1 t❡♠♦s q✉❡
−1 n ≤ cosn n ≤ 1 n ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ lim n→∞ −1
n ≤nlim→∞
cosn
n ≤nlim→∞
1
n✳ ▲♦❣♦ nlim→∞
cosn n = 0✳
❈♦r♦❧ár✐♦ ✷✳✶✳ ❯♠❛ s❡q✉ê♥❝✐❛{an}❝♦♥✈❡r❣❡ ♣❛r❛ ls❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s s✉❜s❡q✉ê♥❝✐❛s
{a2n} ❡ {a2n−1} ❝♦♥✈❡r❣❡♠ ♣❛r❛ l✳
❊①❡♠♣❧♦ ✷✳✶✺✳ ❆ s❡q✉ê♥❝✐❛(2, 0, 2, 0, · · ·)✱ ❝✉❥♦ ♥✲és✐♠♦ t❡r♠♦ éan= 1+(−1)n+1✱ é ❧✐♠✐t❛❞❛ ♠❛s ♥ã♦ é ❝♦♥✈❡r❣❡♥t❡ ♣♦rq✉❡ ♣♦ss✉✐ ❞✉❛s s✉❜s❡q✉ê♥❝✐❛s ❝♦♥st❛♥t❡s✱a2n−1 = 2
❡ a2n = 0 ❝♦♠ ❧✐♠✐t❡s ❞✐st✐♥t♦s✳
❚❡♦r❡♠❛ ✷✳✽✳ ✭❚❡st❡ ❞❛ ❘❛③ã♦ ♣❛r❛ ❙❡q✉ê♥❝✐❛✮ ❙❡ ✉♠❛ s❡q✉ê♥❝✐❛ {an} ❞❡ t❡r♠♦s ♣♦s✐t✐✈♦s s❛t✐s❢❛③ à ❝♦♥❞✐çã♦✿
lim
n→∞
an+1
an
=l < 1,
❡♥tã♦ ❡❧❛ ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✳
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♥ú♠❡r♦ r✱ t❛❧ q✉❡l < r <1 ❡ an+1 an
< r✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝❡rt❛
♦r❞❡♠ n0 ✭❡st❡ í♥❞✐❝❡ n0 ❝♦rr❡s♣♦♥❞❡ à ❡s❝♦❧❤❛ ❞❡ ǫ=r−l✮✳ ❈♦♠♦ r < l✱ t❡♠♦s ♣❛r❛
n ≥n0 q✉❡
0< an+1 < anr < an,
❡✱ ♣♦rt❛♥t♦✱ ❛ s❡q✉ê♥❝✐❛ {an} s❡ t♦r♥❛ ❞❡❝r❡s❝❡♥t❡ ❛ ♣❛rt✐r ❞❛ ♦r❞❡♠ n0✳
❆ss✐♠✱
0< an≤an0, ♣❛r❛ t♦❞♦n≥n0, ✭✷✳✻✮
❡ ❛ r❡❧❛çã♦ ❛❝✐♠❛ ❛ss❡❣✉r❛ ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡q✉ê♥❝✐❛ {an}✱ ❝♦❧♦❝❛♥❞♦✲❛ ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✺✱ s❡♥❞♦ ♣♦r ❝♦♥s❡❣✉✐♥t❡ ❝♦♥✈❡r❣❡♥t❡✳ P❛r❛ ♣r♦✈❛r q✉❡ {an} ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✱ r❛❝✐♦❝✐♥❛♠♦s ♣♦r ❛❜s✉r❞♦ ❛❞♠✐t✐♥❞♦ q✉❡ s❡✉ ❧✐♠✐t❡ é ✉♠ ♥ú♠❡r♦ sN eq0✳
❈♦♠♦ ❛ s❡q✉ê♥❝✐❛{an+1}t❛♠❜é♠ ❝♦♥✈❡r❣❡ ♣❛r❛s✱ ♣♦r s❡r ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ {an}✱ r❡s✉❧t❛ q✉❡✿
l = lim
n→∞
an+1
an
= lim
n→∞an
+1
lim
n→∞an
= s
s = 1,
❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ l <1✳
❊①❡♠♣❧♦ ✷✳✶✻✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❛❝✐♠❛✱ ♣r♦✈❛r❡♠♦s q✉❡ ❛s s❡q✉ê♥❝✐❛s✿
an=
n! nn, bn=
rn
n!, r >0, cn=
n!
1·3·5·. . .·(2n−1) ❡ dn= np
2n.
❈♦♠ ❡❢❡✐t♦✱ t♦❞❛s ❡ss❛s s❡q✉ê♥❝✐❛s sã♦ ❞❡ t❡r♠♦s ♣♦s✐t✐✈♦s✱ ❡ ❞❡ ❛ ❝♦r❞♦ ❝♦♠ ♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ ❡♠ ❝❛❞❛ ❝❛s♦ ♦ ❧✐♠✐t❡ ❞❛ r❛③ã♦ xn+1
xn
< 1✳ P❛r❛ ❛
s❡q✉ê♥❝✐❛ {an}✱ t❡♠♦s✿
lim
n→∞
an+1
an
= lim
n→∞
(n+ 1)! (n+ 1)n+1 ·
nn
n! = limn→∞
n n+ 1
n
= 1
✸✹ ❙❡q✉ê♥❝✐❛s ◆✉♠ér✐❝❛s
❆q✉✐✱ ✉s❛♠♦s ♦ ❧✐♠✐t❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦ ♣❛r❛ ❛ ❢✉♥çã♦✿
f(x) =
1 + 1 x
x
.
P❛r❛ ❛ s❡q✉ê♥❝✐❛ {bn} t❡♠♦s✿
lim
n→∞
bn+1
bn
= lim
n→∞
rn+1
(n+ 1)!· n!
rn =rnlim→∞
1
n+ 1 = 0 <1.
P❛r❛ ❛ s❡q✉ê♥❝✐❛ {cn} t❡♠♦s✿
lim
n→∞
cn+1
cn
= lim
n→∞
1·3·5·. . .·(2n−1)·(n+ 1)!
1·3·5·. . .·(2n−1)·(2n+ 1)·n! = limn→∞
n+ 1 2n+ 1 =
1 2 <1.
❋✐♥❛❧♠❡♥t❡ ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ {dn} t❡♠♦s✿
lim
n→∞
dn+1
dn
= lim
n→∞
(n+ 1)p
2n+1 ·
2n
np =
1 2nlim→∞
n+ 1 n
p
= 1
2 <1.
❉❡✜♥✐çã♦ ✷✳✾✳ ❯♠❛ s❡q✉ê♥❝✐❛ {an}, n∈N é ❞✐t❛ ❞❡ ❈❛✉❝❤② s❡
P❛r❛ t♦❞♦ ǫ >0 ❡①✐st❡ ◆ t❛❧ q✉❡ n, m≥N ✐♠♣❧✐❝❛ q✉❡ |an−am|< ǫ. ❯♠❛ s❡q✉ê♥❝✐❛ é ❞❡ ❈❛✉❝❤② s❡ s❡✉s t❡r♠♦s s❡ ❛♣r♦①✐♠❛♠ ✉♥s ❞♦s ♦✉tr♦s✳ ◆ã♦ ❛♣❡♥❛s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ♠❛s s✐♠ t♦❞♦s ♦s s❡✉s t❡r♠♦s✳ ➱ ♥♦r♠❛❧ ❛❝r❡❞✐t❛r q✉❡ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ é ❞❡ ❈❛✉❝❤② ❡ ✈✐❝❡✲✈❡rs❛✳
❚❡♦r❡♠❛ ✷✳✾✳ ✭❙❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②✮ ❯♠❛ s❡q✉ê♥❝✐❛ é ❝♦♥✈❡r❣❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ é ❞❡ ❈❛✉❝❤②✳
❙❡❥❛{an}✉♠❛ s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ❧✐♠✐t❡l✳ ❉❛❞♦ǫ >0✱ ❡①✐st❡N ∈N t❛❧ q✉❡ s❡ n≥N✱ ❡♥tã♦ |an−l|< ǫ2✳ P♦rt❛♥t♦✱ s❡m, n≥N t❡♠♦s
|an−am| ≤ |an−l|+|l−am|<
ǫ
2 +
ǫ 2 =ǫ.
❈♦♥❝❧✉í♠♦s q✉❡{an} é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡{an}é ❞❡ ❈❛✉❝❤②✳ ❯♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦ ❛♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ ♠♦str❛ q✉❡{an}é ❧✐♠✐t❛❞❛ ✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦✲ ❲❡✐❡rstr❛ss✱ {an} t❡♠ s✉❜s❡q✉ê♥❝✐❛ {ank} ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ♦ ❧✐♠✐t❡ l✳
❆❣♦r❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ an → l✳ ❙❡❥❛ ǫ > 0✳ ❈♦♠♦ {an} é ❞❡ ❈❛✉❝❤②✱ ❡①✐st❡
N ∈N t❛❧ q✉❡
m, m ≥ N ✐♠♣❧✐❝❛ q✉❡ |an−am|<
ǫ
2. ✭✷✳✼✮
❈♦♠♦ ank →l✱ ❡①✐st❡ k ∈N t❛❧ q✉❡ nk ≥N ❡|ank −l|< ǫ
2✳ ❉❛í ❡ ❞❡ ✭✷✳✼✮ s❡❣✉❡ q✉❡✱
s❡ n≥N✱ ❡♥tã♦
|an−l| ≤ |an−ank|+|ank −l|<
ǫ 2+
ǫ 2 =ǫ.
▲✐♠✐t❡s ■♥✜♥✐t♦s ✸✺
✷✳✹ ▲✐♠✐t❡s ■♥✜♥✐t♦s
❊①✐st❡♠ s❡q✉ê♥❝✐❛s ❞✐✈❡r❣❡♥t❡s q✉❡ ✏♣♦ss✉❡♠ ❧✐♠✐t❡✑ ✳ ❆ ❢r❛s❡ ❞✐t❛ ❛♥t❡r✐♦r♠❡♥t❡ é ❛♣❡♥❛s ✉♠ ❥♦❣♦ ❞❡ ♣❛❧❛✈r❛s✳ ❆s ❞❡✜♥✐çõ❡s q✉❡ ❞❛r❡♠♦s ❛ s❡❣✉✐r ❞✐③ q✉❡ ❝❡rt❛s s❡q✉ê♥❝✐❛s tê♠ ❧✐♠✐t❡s q✉❡ ♥ã♦ sã♦ ♥ú♠❡r♦s r❡❛✐s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♥ã♦ ❞✐r❡♠♦s q✉❡ t❛✐s s❡q✉ê♥❝✐❛s sã♦ ❝♦♥✈❡r❣❡♥t❡s✳
❉❡✜♥✐çã♦ ✷✳✶✵✳ ❙❡❥❛ {an} ✉♠❛ s❡q✉ê♥❝✐❛✳ ❉✐③✲s❡ q✉❡ an t❡♥❞❡ ❛ ♠❛✐s ✐♥✜♥✐t♦ q✉❛♥❞♦nt❡♥❞❡ ❛ ♠❛✐s ✐♥✜♥✐t♦ ♦✉ q✉❡ ♠❛✐s ✐♥✜♥✐t♦ é ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ ❡ ❡s❝r❡✈❡♠♦s an →+∞ ♦✉ lim
n→∞an= +∞ s❡✱
Dado M ∈R, ❡①✐st❡ N ∈N t❛❧ q✉❡ n ≥N ✐♠♣❧✐❝❛ q✉❡ an > M.
❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛ {an} ✉♠❛ s❡q✉ê♥❝✐❛✳ ❉✐③✲s❡ q✉❡ an t❡♥❞❡ ❛ ♠❡♥♦s ✐♥✜♥✐t♦ q✉❛♥❞♦ n t❡♥❞❡ ❛ ♠❡♥♦s ✐♥✜♥✐t♦ ♦✉ q✉❡ ♠❡♥♦s ✐♥✜♥✐t♦ é ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ ❡ ❡s✲
❝r❡✈❡♠♦s an→ −∞ ♦✉ lim
n→∞an =−∞ s❡✱
P❛r❛ t♦❞♦M ∈R, ∃N ∈N t❛❧ q✉❡ n ≥N ✐♠♣❧✐❝❛ q✉❡ an < M.
