❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ❚❡rr❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛✿❯♠❛
❆❜♦r❞❛❣❡♠ P♦ssí✈❡❧ ♣❛r❛ ♦ ❊♥s✐♥♦
❞❡ ❙♦♠❛s ■♥✜♥✐t❛s✳
†♣♦r
❏♦s✐❡❧❞❡s ▼❛rq✉❡s ❞♦s ❙❛♥t♦s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❋❛❣♥❡r ▲❡♠♦s ❞❡ ❙❛♥t❛♥❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊❚✲❯❋❘◆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❉❡③❡♠❜r♦✴✷✵✶✹
◆❛t❛❧ ✲ ❘◆
◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛✿❯♠❛
❆❜♦r❞❛❣❡♠ P♦ssí✈❡❧ ♣❛r❛ ♦ ❊♥s✐♥♦
❞❡ ❙♦♠❛s ■♥✜♥✐t❛s✳
♣♦r
❏♦s✐❡❧❞❡s ▼❛rq✉❡s ❉♦s ❙❛♥t♦s
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊❚✲❯❋❘◆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❙ér✐❡s✳
❆♣r♦✈❛❞♦ ♣♦r✿
Pr♦❢✳ ❉r✳ ❋❛❣♥❡r ▲❡♠♦s ❞❡ ❙❛♥t❛♥❛ ✲ ❯❋❘◆ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳❉r✳ ❋❛❜✐❛♥❛ ❚r✐stã♦ ❞❡ ❙❛♥t❛♥❛ ✲ ❯❋❘◆ ✭▼❡♠❜r♦ ❊①t❡r♥♦✮
Pr♦❢✳ ❉r✳ ❲❛❧t❡r ❇❛t✐st❛ ❉♦s ❙❛♥t♦s ✲ ❯❋● ✭▼❡♠❜r♦ ❊①t❡r♥♦✮
❉❡❞✐❝❛tór✐❛
❆❣r❛❞❡❝✐♠❡♥t♦s
■♥✐❝✐♦ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣♦r ❉❡✉s✱ ♣♦r ♠❡ ❛♠♣❛r❛r ♥❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♣♦r ♠❡ ❞á ❢♦rç❛ ♣❛r❛ s❡❣✉✐r s❡♠♣r❡ ❡♠ ❢r❡♥t❡✱ ❡ ♣♦r ❝♦❧♦❝❛r ♥♦ ♠❡✉ ❝❛♠✐♥❤♦ ♣❡ss♦❛s tã♦ ❡s♣❡❝✐❛✐s✱ s❡♠ ❛s q✉❛✐s ❝❡rt❛♠❡♥t❡ ♥ã♦ t❡r✐❛ ❞❛❞♦ ❝♦♥t❛✦
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢✳ ❉r✳ ❋❛❣♥❡r ▲❡♠♦s ❞❡ ❙❛♥t❛♥❛✱ ♣♦r ❡stá s❡♠♣r❡ ❞✐s♣♦♥í✈❡❧✱ ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ t❛♠❜é♠ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ✐♥t❡❣r❛❧♠❡♥t❡ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❝❡ss♦✳ ❙❡♠ t❛❧ ❛❥✉❞❛ ❡ss❡ tr❛❜❛❧❤♦ ♥ã♦ t❡r✐❛ ❛ ♠❡s♠❛ q✉❛❧✐❞❛❞❡✳
❆♦s ♠❡✉s ♣❛✐s✱ ❏♦sé P❛✉❧♦ ❞♦s ❙❛♥t♦s ❡ ●✐❧✈❛♥❞❛ ▼❛rq✉❡s ❞♦s ❙❛♥t♦s✱ ♠❡✉ ✐♥✜♥✐t♦ ❛❣r❛❞❡❝✐♠❡♥t♦✳ ❙❡♠♣r❡ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠✐♠✱ ❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛ ❞❛r ♦ ♠❡✉ ♠❡❧❤♦r ❡ ❛ ♥✉♥❝❛ ❞❡s✐st✐r ❞♦s ♠❡✉s ♦❜❥❡t✐✈♦s✳
❆ ♠✐♥❤❛ ✐r♠ã✱ ❏â♠✐③❛✱ ♣♦✐s✱ ❛♦ s❡✉ ♠♦❞♦✱ s❡♠♣r❡ s❡ ♦r❣✉❧❤♦✉ ❞❡ ♠✐♠✱ ❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧ ♠❡s♠♦ q✉❛♥❞♦ ♥❡♠ ❡✉ ♠❡s♠♦ ❛❝r❡❞✐t❛✈❛✳
◗✉❡r✐❛ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ❛♠✐❣♦s✱ ♦✉ ♠❡❧❤♦r✱ ✐r♠ã♦s✱ ❏❛♠❡rs♦♥ ❋❡r♥❛♥❞♦ ❡ ❇r✉♥♦ ❚❤✐❛❣♦✱ ♣❡❧❛ ❛❥✉❞❛ ❞❛❞❛ ♥❛s ❤♦r❛s ❞❡ s✉❢♦❝♦✱ ♣❡❧❛ ❣❡♥❡r♦s✐❞❛❞❡ ❡♠ ❝♦♠✲ ♣❛rt✐❧❤❛r ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡ ♣♦r ♥ã♦ ♠❡ ❞❡✐①❛r❡♠ ❢r❛q✉❡❥❛r ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❞♦✐s ❛♥♦s✳ ◆ã♦ ♣♦❞❡r✐❛ ❡sq✉❡❝❡r ❞♦s ❞❡♠❛✐s ❝♦❧❡❣❛s ❞❡ P❘❖❋▼❆❚✿ ❆♥tô♥✐♦ ❘♦❜❡rt♦✱ ▼❛r✲ ❝✐♦✱ ❆❧♠✐r✱ ❘♦sâ♥❣❡❧❛✱ ▼❛r❝♦ ▲✐r❛✱ ▼❛r❝❡❧♦✱ ❘♦❜❡rt♦ ❋❛❣♥❡r ❡ ❱❡♥í❝✐♦✳ ❖❜r✐❣❛❞♦ ♣❡❧❛ ❛t❡♥çã♦ ❡ ♣❡❧❛ ❢♦rç❛ ❞❛❞❛s ❛tr❛✈és ❞♦ ❝♦♠♣❛♥❤❡r✐s♠♦✱ s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♠✐♥✐str❛r❛♠ ❛✉❧❛s ♥♦ P❘❖❋▼❆❚ ♥❛ ❯❋❘◆✱ ♣❡❧❛ ❞❡✲ ❞✐❝❛çã♦ ❞❡st✐♥❛❞❛ ❛♦ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ♥❛s ❞✐s❝✐♣❧✐♥❛s ♠✐♥✐str❛❞❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❘♦♥❛❧❞♦ ❡ ❉é❜♦r❛ q✉❡ ❞❡ ❢❛t♦ ❞❡r❛♠ ❝♦♥tr✐❜✉✐çõ❡s ❡①tr❡♠❛♠❡♥t❡
s✐❣♥✐✜❝❛t✐✈❛s ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝♦♠♦ ♠❛t❡♠át✐❝♦✳
✏ ◆❡♥❤✉♠ ♦✉tr♦ ♣r♦❜❧❡♠❛ ❛❢❡t♦✉ tã♦ ♣r♦❢✉♥❞❛♠❡♥t❡ ♦ ❡s♣ír✐t♦ ❞♦ ❤♦♠❡♠❀ ♥❡♥❤✉♠❛ ♦✉tr❛ ✐❞❡✐❛ tã♦ ❢❡rt✐❧♠❡♥t❡ ❡st✐♠✉❧♦✉ s❡✉ ✐♥t❡❧❡❝t♦❀ ♥❡♥❤✉♠ ♦✉tr♦ ❝♦♥❝❡✐t♦ ♥❡❝❡ss✐t❛ ❞❡ ♠❛✐♦r ❡s❝❧❛r❡❝✐♠❡♥t♦ ❞♦ q✉❡ ♦ ✐♥✜♥✐t♦✑
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ♦ tr❛t❛❞♦ ❞❛ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛✱ ♦ q✉❛❧ tr❛t❛ ❞♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ♣❛rá❜♦❧❛ q✉❡ ❢♦✐ ❢❡✐t♦ ♣♦r ❆rq✉✐♠❡❞❡s✳ P❛r❛ ✐ss♦✱ sã♦ ♥❡❝❡ssár✐❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s✱ ❝♦♠ ♦s q✉❛✐s ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ❛ ✐❞❡✐❛ ❞❡ ♣r♦❝❡ss♦s ✐♥✜♥✐t♦s ✭♦✉ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥✜♥✐t♦✮ ♣❛r❛ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳
P❛❧❛✈r❛s ❝❤❛✈❡✿ ■♥✜♥✐t♦✱ sér✐❡s✱ q✉❛❞r❛túr❛ ❞❛ ♣❛rá❜♦❧❛✱ ❆rq✉✐♠❡❞❡s
❆❜str❛❝t
❚❤✐s ❞✐ss❡rt❛t✐♦♥ ♣r❡s❡♥ts t❤❡ q✉❛❞r❛t✉r❡ ♦❢ t❤❡ ♣❛r❛❜♦❧❛ tr❡❛t②✱ ✇❤✐❝❤ ❞❡❛❧s ✇✐t❤ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❛ ♣❛r❛❜♦❧✐❝ s❡❣♠❡♥t✱✇❤✐❝❤ ✇❛s ♠❛❞❡ ❜② ❆r❝❤✐♠❡❞❡s✳ ❋♦r t❤✐s✱ ❝♦♥s✐❞❡r❛t✐♦♥s ♦♥ s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s ❛r❡ ♥❡❝❡ss❛r②✱ ✇✐t❤ ✇❤✐❝❤ ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ t❤❡ ✐❞❡❛ ♦❢ ✐♥✜♥✐t❡ ♣r♦❝❡ss❡s ✭♦r t❤❡ ❝♦♥❝❡♣t ♦❢ ✐♥✜♥✐t②✮ ❢♦r ❡❧❡♠❡♥t❛r② s❝❤♦♦❧ st✉❞❡♥ts✳
❑❡②✇♦r❞s✿ ■♥✜♥✐t②✱ s❡r✐❡s✱ q✉❛❞r❛t✉r❡ ♦❢ t❤❡ ♣❛r❛❜♦❧❛✱ ❆r❝❤✐♠❡❞❡s
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❙❡q✉ê♥❝✐❛ ❞❡ ◆ú♠❡r♦s ❘❡❛✐s ❡ s❡✉s ▲✐♠✐t❡s ✹
✷✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ◆ú♠❡r♦s ❘❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ❆ ✐❞❡✐❛ ❞❡ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❙❡q✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✸ ❙ér✐❡s ✶✷
✸✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✷ ❙♦♠❛s P❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✸ ❙♦♠❛ ❞♦s t❡r♠♦s ❞❡ ✉♠❛ ❙ér✐❡ ❣❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✹ ◆♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ❙ér✐❡s ❝♦♥✈❡r❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✺ ❙ér✐❡ ❍❛r♠ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✻ ❍✐stór✐❛ ❞♦ ①❛❞r❡③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✹ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ✶✾
✹✳✶ ❊✉❞ó①✐♦ ❡ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✹✳✷ ❱♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✺ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛ ✷✼
✺✳✶ ❈♦♥❝❡✐t♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✺✳✷ P❛rá❜♦❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✸ ❚r✐â♥❣✉❧♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✺✳✹ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛ ❡ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✺ ▼ét♦❞♦ ❞❡ ❉✉♣❧❛ r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✺
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❊①❡♠♣❧♦s ❞❡ s♦♠❛s ✐♥✜♥✐t❛s s✉r❣❡♠ ♠✉✐t♦ ❝❡❞♦✱ ❛✐♥❞❛ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♥♦ ❡st✉❞♦ ❞❡ ❞í③✐♠❛s ♣❡r✐ó❞✐❝❛s✳ ❈♦♠ ❡❢❡✐t♦✱ ✉♠❛ ❞í③✐♠❛ ❝♦♠♦0,555. . . ♥❛❞❛ ♠❛✐s
é ❞♦ q✉❡ ✉♠❛ sér✐❡ ❣❡♦♠étr✐❝❛ ✐♥✜♥✐t❛✳ ❱❡❥❛✿
0,555· · ·= 5×0,111· · ·= 5
1 10+
1 100 +
1
1000 +. . .
= 5
1 10+
1 102 +
1
103 +. . .
= 5
1 10
1− 1 10
= 5 9.
❙❡♥❞♦ ❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡ ♦❜t✐❞❛ ❞❛ ❢ór♠✉❧❛ ❞❛ s♦♠❛ ❞♦s t❡r♠♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✐♥✜♥✐t❛✳ ◗✉❛♥❞♦ s❡ ❡♥s✐♥❛ ❡ss❛s ❞í③✐♠❛s✱ ✉s❛✲s❡ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦✱ ♥♦ q✉❛❧ ❛ s♦♠❛ ✐♥✜♥✐t❛ ♥ã♦ ❛♣❛r❡❝❡ ❡①♣❧✐❝✐t❛♠❡♥t❡✿
x= 0,555· · · ⇒10x= 5,555· · ·= 5 + 0,555· · ·= 5 +x⇒9x= 5⇒x= 5 9
◆♦ ❡♥s✐♥♦ ♠é❞✐♦ ✈♦❧t❛✲s❡ ❛ ❢❛❧❛r s♦❜r❡ s♦♠❛s ✐♥✜♥✐t❛s✱ s❡♥❞♦ q✉❡ ❛❣♦r❛✱ ❛♣ós ❢♦r♠❛❧✐③❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s✱ é ❛♣r❡s❡♥t❛❞❛ ❛♦ ❛❧✉♥♦ ✉♠❛ ❢ór✲ ♠✉❧❛ q✉❡ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞❡ ✐♥✜♥✐t♦s t❡r♠♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ q✉❛♥❞♦ ❛ r❛③ã♦ q é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡♥tr❡ −1 ❡ 1✳ ❆ ✐❞❡✐❛ ✐♥❣é♥✉❛ ❡ ♥ã♦ ❝rít✐❝❛
❞❡ s♦♠❛ ✐♥✜♥✐t❛ ♥ã♦ ❝♦st✉♠❛ ♣❡rt✉r❜❛r ♦ ❡st✉❞❛♥t❡ q✉❡ ❧♦❣♦ ❛ss♦❝✐❛ ❛ ❛❞✐çã♦ ❝♦♠ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ t❡r♠♦s✳ ❆♦ ❛♥❛❧✐s❛r ♦s t❡r♠♦s ✉♠ ❛ ✉♠✱ ❡❧❡ ♣❡r❝❡❜❡ q✉❡ ❡ss❡s
✈ã♦ ✜❝❛♥❞♦ ♠❡♥♦r❡s ❡ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦s ❞❡ ③❡r♦✱ ♦ q✉❡ ❧❡✈❛ ❛ ✐♥t❡r♣r❡t❛r q✉❡ ❛ s♦♠❛ ❞❡ss❡s t❡r♠♦s ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝❡rt❛ q✉❛♥t✐❞❛❞❡ ♥ã♦ ✐♥t❡r❢❡r❡ ♠✉✐t♦ ♥♦ r❡s✉❧t❛❞♦ ✜♥❛❧✱ q✉❡ ♣♦r s✉❛ ✈❡③✱ s❡ ❛♣r♦①✐♠❛ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❖ ♣r♦❜❧❡♠❛ é q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ só ❢❛③ s❡♥t✐❞♦ q✉❛♥❞♦ ❛♣❧✐❝❛❞❛ ❛ ✉♠ ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ P♦ré♠✱ ❞❡✈✐❞♦ à ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❡♠ R✱ ♣♦❞❡♠♦s ❡❢❡t✉❛r
✉♠❛ s♦♠❛ ❞❡ 3,4,5, . . . ,100 ♦✉ ♠❛✐s ♥ú♠❡r♦s✱ s❡♠ ✐♥❝♦rr❡r ❡♠ ❡rr♦s✳ P♦r ❡①❡♠♣❧♦✱
♣♦❞❡♠♦s ♦❜t❡r ❛ s♦♠❛ 2 + 4 + 8 ❝♦♠♦ 2 + 4 + 8 = (2 + 4) + 8 ✱ ♦✉ ❡♥tã♦ ❝♦♠♦ 2 + 4 + 8 = 2 + (4 + 8)✱ ♦ r❡s✉❧t❛❞♦ é ♦ ♠❡s♠♦✳ P♦ré♠✱ ❡♥❝❛r❛r s♦♠❛s ✐♥✜♥✐t❛s
♥♦s ♠❡s♠♦s ♠♦❧❞❡s ❞❛s s♦♠❛s ✜♥✐t❛s✱ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s✱ ♣♦❞❡ ♥♦s ❧❡✈❛r ❛ ❞✐✜❝✉❧❞❛❞❡s ❡ ❝♦♥❝❧✉sõ❡s ❡q✉✐✈♦❝❛❞❛s✳ ❈♦♠♦ ❜❡♠ ✐❧✉str❛ ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s✱ ❞❛❞♦ ♣❡❧❛ ❝❤❛♠❛❞❛ sér✐❡ ❞❡ ❣r❛♥❞✐✳
S= 1−1 + 1−1 + 1−1 +· · ·=
+P∞
n=0
(−1)n
❯t✐❧✐③❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❞❡ ❢♦r♠❛ ❝♦♥✈❡♥✐❡♥t❡ ♣♦❞❡♠♦s ♦❜t❡r ♦s s❡❣✉✐♥✲ t❡s r❡s✉❧t❛❞♦s✿
❛✮S = (1−1) + (1−1) + (1−1) + (1−1) +· · ·= 0
❜✮S = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) +· · ·= 1
❝✮S = 1−(1−1 + 1−1 + 1−1 +. . .)⇒S = 1−S ⇒2S = 1 ⇒S= 1/2
❈♦♠♦ ❞❡❝✐❞✐r ❡♥tã♦ ♦ r❡s✉❧t❛❞♦❄ S = 0✱ 1♦✉ 1/2❄ ❙❡ ♦ ❧❡✐t♦r t❡♠ ✉♠❛ ❝♦♠♣r❡✲
❡♥sã♦ ❜❡♠ ❡st❛❜❡❧❡❝✐❞❛ s♦❜r❡ sér✐❡s ❝♦♥❝❧✉✐r✐❛ ❢❛❝✐❧♠❡♥t❡ q✉❡ ♥❡♥❤✉♠ ❞♦s r❡s✉❧t❛❞♦s ♣♦❞❡r✐❛ ❡st❛r ❝♦rr❡t♦ ✉♠❛ ✈❡③ q✉❡ ❡ss❛ s♦♠❛ tr❛t❛✲s❡ ❞❡ ✉♠❛ sér✐❡ ❞✐✈❡r❣❡♥t❡✳
P❡r❝❡❜❡♠♦s ❛ss✐♠ q✉❡ ♦ ♣r♦❢❡ss♦r ❞❡✈❡ t♦♠❛r ✉♠ ❝❡rt♦ ❝✉✐❞❛❞♦ ❛♦ tr❛❜❛❧❤❛r ❝♦♠ ❡ss❡ t❡♠❛✱ ♣♦✐s✱ é ♦ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ q✉❡ ♦ ❛❧✉♥♦ t❡♠ ❝♦♠ ❛s ✐❞❡✐❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ✐♥✜♥✐t♦✳ ❖ ♣r♦❜❧❡♠❛ é q✉❡ ♠❡s♠♦ ♦s ❧✐❝❡♥❝✐❛❞♦s ❡♠ ♠❛t❡♠át✐❝❛✱ q✉❡ sã♦ ♦s ♣r♦❢❡ss♦r❡s ❞❡st❡s ❛❧✉♥♦s✱ r❡✈❡❧❛♠ ❡♠ s✉❛ ♠❛✐♦r✐❛ ❝❡rt❛ ✐♥❛❜✐❧✐❞❛❞❡ ❡♠ tr❛❜❛❧❤❛r ❝♦♠ s♦♠❛s ✐♥✜♥✐t❛s✱ s❡ ❧✐♠✐t❛♥❞♦ ❛ ♠❡r❛ r❡♣r♦❞✉çã♦ ❞♦ q✉❡ ♦s ❧✐✈r♦s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ tr❛③❡♠ s♦❜r❡ ❡ss❡ ❛ss✉♥t♦✳
❊ss❛ ❞✐ss❡rt❛çã♦ ❛ss✐♠ s❡ ❥✉st✐✜❝❛ ❝♦♠♦ ✉♠ ❛❣❡♥t❡ ♣r♦✈♦❝❛❞♦r ❞❡ss❛ t❡♠át✐❝❛✱ t❡♥❞♦ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❞♦ ❧❡✐t♦r às ✈ár✐❛s ♣❡rs♣❡❝t✐✈❛s ♣❡❧❛s q✉❛✐s ❛s s♦♠❛s ✐♥✜♥✐t❛s ♣♦❞❡♠ s❡r ❛✈❛❧✐❛❞❛s ❡ ❝♦♠♦ ❢♦r❛♠ ❡♥❝❛r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ♥♦s ✈❛❧❡r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦✱ ❛ q✉❛❞rát✉r❛ ❞❛ ♣❛rá❜♦❧❛✱ ✉♠ ❞♦s ♣r✐♠❡✐r♦s ♣r♦❜❧❡♠❛s s♦❜r❡ s♦♠❛s ✐♥✜♥✐t❛s✱ q✉❡ ❝✉r✐♦s❛♠❡♥t❡✱ ♣❛r❡❝❡ s❡r ♣♦✉❝♦ ❝♦♥❤❡❝✐❞♦ ♥♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ❛❧é♠ ❞❡ ❡st❡♥❞❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛❞✐çã♦ ♣❛r❛ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ❡ ❞❡✜♥✐r ♦ q✉❡ s✐❣♥✐✜❝❛ t❛❧ s♦♠❛✳ ❈❤❛♠❛r❡♠♦s ❡st❛s ✏s♦♠❛s ✐♥✜♥✐t❛s✑ ❞❡ sér✐❡s✳
❈❛♣ít✉❧♦ ✷
❙❡q✉ê♥❝✐❛ ❞❡ ◆ú♠❡r♦s ❘❡❛✐s ❡ s❡✉s
▲✐♠✐t❡s
❖ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ é ♦ ♠❛✐s ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ♣♦✐s é ♥❡❧❡ q✉❡ s❡ ❜❛s❡✐❛♠ ♥❛ ▼❛t❡♠át✐❝❛ ❛t✉❛❧ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❞✐✈❡r❣ê♥❝✐❛✱ ❝♦♥t✐♥✉✐❞❛❞❡✱ ❞❡r✐✈❛❞❛ ❡ ✐♥t❡❣r❛❧✳
❊♠❜♦r❛ ❢✉♥❞❛♠❡♥t❛❧✱ ❡ss❡ ❝♦♥❝❡✐t♦ ❞❡♠♦r♦✉ ♠❛✐s ❞❡ ❞♦✐s ♠✐❧ê♥✐♦s ♣❛r❛ ✜♥❛❧✲ ♠❡♥t❡ s❡r r✐❣♦r♦s❛♠❡♥t❡ ❞❡✜♥✐❞♦ ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❞♦ sé❝✉❧♦ ❳■❳✳ ◆❡ss❡ ❝❛♣ít✉❧♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♥♦çã♦ ❞❡ ❧✐♠✐t❡ s♦❜ s✉❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✱ ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ❡st❡ ❛ss✉♥t♦ r❡❝♦♠❡♥❞❛♠♦s ❬✾❪✳
✷✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ◆ú♠❡r♦s ❘❡❛✐s
❆ ✐❞❡✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s é ❛ ❞❡ ❡s❝♦❧❤❡r ♥ú♠❡r♦s r❡❛✐s ❡ ❝♦❧♦❝á✲❧♦s ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♦r❞❡♠ ✭❝♦♠ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ r❡♣❡t✐çã♦✮✱ ♦✉ s❡❥❛✱ s❛❜❡♠♦s q✉❛❧ ❞❡❧❡s é ♦ ♣r✐♠❡✐r♦✱ ♦ s❡❣✉♥❞♦✱ ❡t❝✳ ❊ss❛ ✐❞❡✐❛ é r❡♣r❡s❡♥t❛❞❛ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♥❛ ❞❡✜♥✐çã♦ ❛❜❛✐①♦✿
❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭❈♦♥❝❡✐t♦ ❞❡ s❡q✉ê♥❝✐❛✮ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛
✷✳✶✳ ❙❊◗❯✃◆❈■❆ ❉❊ ◆Ú▼❊❘❖❙ ❘❊❆■❙
❢✉♥çã♦ x:N→R q✉❡ ❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❛ss♦❝✐❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ xn=x(n)✱
❝❤❛♠❛❞♦ ♦ ♥✲és✐♠♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✳
❊s❝r❡✈❡✲s❡ (x1, x2, x3, x4, . . . , xn, . . .)♦✉(xn)n∈N✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡(xn)✱ ♣❛r❛ ✐♥✲
❞✐❝❛r ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ ♥✲és✐♠♦ t❡r♠♦ é xn✳ ❬✾❪
◆ã♦ ❝♦♥❢✉♥❞❛ ❛ s❡q✉ê♥❝✐❛ (xn) ❝♦♠ ♦ ❝♦♥❥✉♥t♦ {x1, x2, x3, x4, . . . , xn, . . .} ❞♦s s❡✉s t❡r♠♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛ s❡q✉ê♥❝✐❛(1,1, . . . ,1, . . .)♥ã♦ é ♦ ♠❡s♠♦ q✉❡ ♦ ❝♦♥❥✉♥t♦
{1}✳ ❖✉ ❡♥tã♦✿ ❛s s❡q✉ê♥❝✐❛s (0,1,0,1, . . .) ❡ (0,0,1,0,0,1, . . .) sã♦ ❞✐❢❡r❡♥t❡s ♠❛s
♦ ❝♦♥❥✉♥t♦ ❞♦s s❡✉s t❡r♠♦s é ♦ ♠❡s♠♦✱ ✐❣✉❛❧ ❛ {0,1}✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s t❡r♠♦s ❞❡
✉♠❛ s❡q✉ê♥❝✐❛ (xn) é ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦x:N→R✳
❊①❡♠♣❧♦ ✷✳✶✳✶ ❆ s❡q✉ê♥❝✐❛ (1,2,1,2,1,2, . . .) ❝♦rr❡s♣♦♥❞❡ à ❢✉♥çã♦ x(n) = 1 s❡
n é ✐♠♣❛r ❡x(n) = 2s❡ n é ♣❛r❀ ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s t❡r♠♦s é ♦ ❝♦♥❥✉♥t♦X ={1,2}✳
❊♠ ❣❡r❛❧✱ ❝❤❛♠❛r❡♠♦s ❞❡ ❝♦♥st❛♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛(xn)t❛❧ q✉❡xn=k✱∀n∈N✱ ♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡ ✜①❛❞❛✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ s❡q✉ê♥❝✐❛ é ❝♦♥st❛♥t❡
q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡✉s t❡r♠♦s é ✉♥✐tár✐♦✳
❆s s❡q✉ê♥❝✐❛s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ❝♦♠♦ ❧✐♠✐t❛❞❛s ♦✉ ✐❧✐♠✐t❛❞❛s✳ ❆s s❡q✉ê♥✲ ❝✐❛s ❧✐♠✐t❛❞❛s ♣♦❞❡♠ s❡r s✉❜❞✐✈✐❞✐❞❛s ❡♠ s❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛s s✉♣❡r✐♦r♠❡♥t❡ ❡ ❧✐♠✐t❛❞❛s ✐♥❢❡r✐♦r♠❡♥t❡✳ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) ❞✐③✲s❡ ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐♥❢❡r✐♦r♠❡♥t❡✮ q✉❛♥❞♦ ❡①✐st❡ c ∈ R t❛❧ q✉❡ xn ≤ c ✭r❡s♣❡❝t✐✈❛✲
♠❡♥t❡ xn≥c✮ ♣❛r❛ t♦❞♦ n ∈N✳ ❉✐③✲s❡ q✉❡ ❛ s❡q✉ê♥❝✐❛(xn)é ❧✐♠✐t❛❞❛ q✉❛♥❞♦ ❡❧❛ é ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ❡ ✐♥❢❡r✐♦r♠❡♥t❡✳ ■st♦ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ ❡①✐st❡ k >0 t❛❧
q✉❡|xn| ≤k ♣❛r❛ t♦❞♦n∈N✳ ◗✉❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ❧✐♠✐t❛❞❛ ✭✐♥❢❡r✐♦r♠❡♥t❡ ♦✉ s✉♣❡r✐♦r♠❡♥t❡✮ ❞✐r❡♠♦s q✉❡ é ✐❧✐♠✐t❛❞❛ ✭✐♥❢❡r✐♦r♠❡♥t❡ ♦✉ s✉♣❡r✐♦r♠❡♥t❡✮✳
❊①❡♠♣❧♦ ✷✳✶✳✷ ❙❡ a > 1 ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ (a, a2
, . . . , an, . . .) é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r✲ ♠❡♥t❡ ♣♦ré♠ ♥ã♦ s✉♣❡r✐♦r♠❡♥t❡✳ P❛r❛ ✈❡r✐✜❝❛r q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ é ✐❧✐♠✐t❛❞❛ s✉♣❡✲ r✐♦r♠❡♥t❡✱ ✈❛♠♦s ✉s❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇❡r♥♦✉❧❧✐✱ ❛ q✉❛❧ ❞✐③ q✉❡ (1 +x)n>1+nx✱
∀n ∈ N ❡ ∀x >−1✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ ❡ ♣♦❞❡
✷✳✶✳ ❙❊◗❯✃◆❈■❆ ❉❊ ◆Ú▼❊❘❖❙ ❘❊❆■❙
s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✾❪ ✳ ❈♦♠♦ a > 1✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r a = 1 +x✱ ❝♦♠ x > 0
❆ss✐♠✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇❡r♥♦✉❧❧✐✱ t❡♠♦s an = (1 +x)n
>1 +nx✳ ❉❛❞♦ k ∈R
❝♦♠ k > 0✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ n0 >
k
x ✶✱ ❧♦❣♦ n0x > k ⇒ 1 +nox > k ⇒ an0
= (1 +x)n0
✳ ❘❡s✉♠✐♥❞♦✱ ♣r♦✈❛♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ k >0✱ ❡①✐st❡
n0 ∈ N❀ an0 > k✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❛ s❡q✉ê♥❝✐❛ an é ✐❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡✳ P♦r
♦✉tr♦ ❧❛❞♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ 1< a ♣♦r an ♦❜t❡♠♦s
an< an+1✳ ❙❡❣✉❡✲s❡ q✉❡
a < an ♣❛r❛ t♦❞♦n ∈N✳ ▲♦❣♦(an)é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r a✳
❊①❡♠♣❧♦ ✷✳✶✳✸ ❆ s❡q✉ê♥❝✐❛
1 2,
2 3, . . . ,
n
n+ 1, . . . ,
é ❧✐♠✐t❛❞❛ ♣♦✐san=
n n+ 1 = 1
1 + 1
n
é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r 0 ❡ s✉♣❡r✐♦r♠❡♥t❡ ♣♦r 1✳
❉❡✜♥✐çã♦ ✷✳✶✳✷ ✭❙❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡✮ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) s❡rá ❞✐t❛ ❝r❡s❝❡♥t❡ s❡ xn < xn+1 ♣❛r❛ t♦❞♦ n ∈ N✳ ❉✐r❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ é ♥ã♦ ❞❡❝r❡s❝❡♥t❡ s❡
xn+1 ≥xn ♣❛r❛ t♦❞♦n ∈N✳
❉❡✜♥✐çã♦ ✷✳✶✳✸ ✭❙❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡✮ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) s❡rá ❞✐t❛ ❞❡❝r❡s✲ ❝❡♥t❡ s❡ xn+1 < xn ♣❛r❛ t♦❞♦ n ∈ N✳ ❉✐r❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ é ♥ã♦ ❝r❡s❝❡♥t❡✱ s❡
xn+1 ≤xn ♣❛r❛ t♦❞♦n ∈N✳
❊①❡♠♣❧♦ ✷✳✶✳✹ ❆ s❡q✉ê♥❝✐❛ (1,2,3,4, . . . , n, . . .) é ✉♠❛ s❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡ ♣♦✐s
n = xn < xn+1 = n + 1 ❥á ❛ s❡q✉ê♥❝✐❛ 1, 1 4,
1 16, . . . ,
1 4n−1, . . .
é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ♣♦✐s xn =
1 4n−1 >
1 4n+1.4−1 =
1
4n+1−1 =xn+1✳
❆s s❡q✉ê♥❝✐❛s ❝r❡s❝❡♥t❡s✱ ♥ã♦✲❞❡❝r❡s❝❡♥t❡s✱ ❞❡❝r❡s❝❡♥t❡s ♦✉ ♥ã♦✲❝r❡s❝❡♥t❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ s❡q✉ê♥❝✐❛s ♠♦♥ót♦♥❛s✳
✶ ✉s❛♠♦s ❛q✉✐ ✉♠ ❢❛t♦ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❛ s❛❜❡r✱ q✉❡Né
✉♠ ❝♦♥❥✉♥t♦ ✐❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✱ ♦✉ s❡❥❛✱ ❞❛❞♦ A ∈ R ❝♦♠ A > 0 ✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡
n0> A✳
✷✳✷✳ ❆ ■❉❊■❆ ❉❊ ▲■▼■❚❊ ❉❊ ❯▼❆ ❙❊◗❯✃◆❈■❆
❊①❡♠♣❧♦ ✷✳✶✳✺ ❙❡❥❛ a ∈ R✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ (a, a2
, a3
, a4
, . . . , an, . . . ,) ❞❛s ♣♦tê♥❝✐❛s ❞❡ a✱ ❝♦♠ ❡①♣♦❡♥t❡ n ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❙❡ a = 0 ♦✉ a = 1✱ t❡♠✲s❡
❡✈✐❞❡♥t❡♠❡♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥st❛♥t❡✳ ❙❡ 0< a <1✱ ❛ s❡q✉ê♥❝✐❛ é ❞❡❝r❡s❝❡♥t❡ ❡
❧✐♠✐t❛❞❛✳ ❈♦♠ ❡❢❡✐t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ a < 1 ♣❡❧♦
♥ú♠❡r♦ ♣♦s✐t✐✈♦ an ♦❜t❡♠♦s an+1
< an✱ ♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝♦♥❧✉sã♦ q✉❡ ❝❛❞❛ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛ é ♠❡♥♦r ❞♦ q✉❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r✱ ❧♦❣♦ ❛ s❡q✉ê♥❝✐❛ é ❞❡❝r❡s❝❡♥t❡✳ ❈♦♠♦ t♦❞♦s ♦s s❡✉s t❡r♠♦s sã♦ ♣♦s✐t✐✈♦s t❡♠♦s 0 < an < 1 ♣❛r❛ t♦❞♦ n✳ ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ♦ ❝❛s♦−1< a < 0✳ ❊♥tã♦ ❛ s❡q✉ê♥❝✐❛ (an)♥ã♦ é ♠❛✐s ♠♦♥ót♦♥❛ ✭s❡✉s t❡r♠♦s sã♦ ❛❧t❡r♥❛❞❛♠❡♥t❡ ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✮ ♠❛s ❛✐♥❞❛ é ❧✐♠✐t❛❞❛ ♣♦✐s |an|=|a|n✱ ❝♦♠
0<|a|<1✳ ❖ ❝❛s♦ a=−1 é tr✐✈✐❛❧❀ ❛ s❡q✉ê♥❝✐❛ (an) é (−1,1,−1,1, . . .)✳ ◗✉❛♥❞♦
a >1 ❖❜t❡♠✲s❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ q✉❛♥❞♦ a <−1✱ ❛ s❡q✉ê♥❝✐❛ (an) ♥ã♦ é ♠♦♥ót♦♥❛ ✭♣♦✐s s❡✉s t❡r♠♦s sã♦ ❛❧t❡r♥❛❞❛♠❡♥t❡ ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✮ ❡ é ✐❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ❡ ✐♥❢❡r✐♦r♠❡♥t❡✳
✷✳✷ ❆ ✐❞❡✐❛ ❞❡ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❙❡q✉ê♥❝✐❛
❱❛♠♦s ❛♥❛❧✐s❛r ❛ s❡q✉ê♥❝✐❛ 1 2n
✳ ➱ ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡ ❛ s❡q✉ê♥❝✐❛ é ❞❡❝r❡s❝❡♥t❡ ❝♦♠ t♦❞♦s ♦s s❡✉s t❡r♠♦s ♣♦s✐t✐✈♦s ✱ ♦✉ s❡❥❛✱❞❛❞♦ n > m ✱ t❡♠✲s❡ q✉❡ 0< 1
2n <
1 2m✳
❈♦♥s✐❞❡r❡♠♦s✱ ❛❣♦r❛✱ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ❝❡♥tr♦ ③❡r♦ ❡ r❛✐♦ ♣❡q✉❡♥♦✱ ❞✐❣❛♠♦s − 1 109,
1 109
✱ q✉❡✱ ❝♦♥✈❡♥❤❛♠♦s✱ é ♠✉✐t♦ ♣❡q✉❡♥♦✳ ❆❣♦r❛✱ ❝♦♠♦ 1
230 = 1
1073741824 < 1 109 <
1 229 = 1
536870912✱ ✈❡♠♦s q✉❡ 1
230 ∈ −
1 109,
1 109
✳ ◆❛ ✈❡r❞❛❞❡✱ ❝♦♠♦ ♣❛r❛ t♦❞♦ n ≥ 30 t❡♠♦s
q✉❡ 1
2n <
1
230✱ ❡♥tã♦
1 2n ∈ −
1 109,
1 109
✳
■ss♦ ♥♦s ♠♦str❛ q✉❡ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ ✈❛❧♦r ❞❡ n✱ ❛ s❛❜❡r✱ n = 30✱ t♦❞♦s ♦s
t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ♣❡rt❡♥❝❡♠ ❛♦ ✐♥t❡r✈❛❧♦ − 1 109,
1 109
✳
▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ♦ q✉❡ ❛✜r♠❛♠♦s ❛❝✐♠❛ ♥ã♦ é r❡str✐t♦ ❛♦ ✐♥t❡r✈❛❧♦ ❡s❝♦❧❤✐❞♦
− 1
109,
1 109
✳ ❉❡ ❢❛t♦✱ ❡s❝♦❧❤❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ r > 0 ❡ ❝♦♥s✐❞❡r❡
♦ ✐♥t❡r✈❛❧♦ (−r, r)✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n0 ≥1 t❛❧ q✉❡ n0 >
1
r✱ ❧♦❣♦
1
n0
< r✳
✷✳✷✳ ❆ ■❉❊■❆ ❉❊ ▲■▼■❚❊ ❉❊ ❯▼❆ ❙❊◗❯✃◆❈■❆
❈♦♠♦ 2n0 > n
0✱ s❡❣✉❡✲s❡ q✉❡ 1 2n0 <
1
n0 < r✳
◆❛ ✈❡r❞❛❞❡✱ ❝♦♠♦ ♣❛r❛ t♦❞♦n > n0 t❡♠✲s❡ q✉❡ 1 2n <
1
2n0✱ ♦❜t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦
n > n0✱ 1 2n < r✳
❱❡♠♦s✱ ♣♦rt❛♥t♦✱ q✉❡ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ ✈❛❧♦r n0 ❞❡ n✱ t♦❞♦s ♦s t❡r♠♦s ❞❛
s❡q✉ê♥❝✐❛ ♣❡rt❡♥❝❡♠ ❛♦ ✐♥t❡r✈❛❧♦ (−r, r)✳ ❈♦♠♦ ♦ ♥ú♠❡r♦ r >0♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦
❛r❜✐tr❛r✐❛♠❡♥t❡✱ ✈❡♠♦s q✉❡ ♥ã♦ ✐♠♣♦rt❛ ♦ q✉ã♦ ♣❡q✉❡♥♦ ❡❧❡ s❡❥❛✱ s❡♠♣r❡ ❡①✐st✐rá✱ ♣❛r❛ ❡ss❛ ❡s❝♦❧❤❛ ❞❡ r✱ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n0 ❛ ♣❛rt✐r ❞♦ q✉❛❧ t♦❞♦s ♦s t❡r♠♦s ❞❛
s❡q✉ê♥❝✐❛ ♣❡rt❡♥❝❡rã♦ ❛♦ ✐♥t❡r✈❛❧♦ (−r, r)✳ ➱ ♥❡ss❡ s❡♥t✐❞♦ q✉❡ ❡♥t❡♥❞❡♠♦s q✉❡ ♦s
t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ s❡ ❛♣r♦①✐♠❛♠ ❞❡ ③❡r♦ q✉❛♥❞♦ n ❝r❡s❝❡✳
❋✐❣✉r❛ ✷✳✶✿ ❉♦✐s ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛ 1 2n
❉❡✜♥✐çã♦ ✷✳✷✳✶ ✭▲✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✮ ❙❡❥❛♠ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú✲ ♠❡r♦s r❡❛✐s ❡l ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❉✐③❡♠♦s q✉❡(xn)❝♦♥✈❡r❣❡ ♣❛r❛l✱ ♦✉ é ❝♦♥✈❡r❣❡♥t❡✱ ❡ ❡s❝r❡✈❡✲s❡ lim
n→∞xn = l✱ q✉❛♥❞♦ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ❝♦♥t❡♥❞♦ l ✭♣♦r
♠❡♥♦r q✉❡ ❡❧❡ s❡❥❛✮ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n0✱ ❞❡ ♠♦❞♦ q✉❡ xn∈I ♣❛r❛ t♦❞♦ n > n0✳
❖❜s❡r✈❛çã♦ ✷✳✷✳✶ ◗✉❛♥❞♦ ♥ã♦ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ l ♣❛r❛ ♦ q✉❛❧ (xn) ❝♦♥✈✐r❥❛✱ ❞✐③❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ (xn) ❞✐✈❡r❣❡✱ ♦✉ q✉❡ é ❞✐✈❡r❣❡♥t❡✳
❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ t♦r♥❛r ♠❛✐s ♦♣❡r❛❝✐♦♥❛❧ ❛ ♥♦ss❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ♥♦t❡ q✉❡✱ ♦ ✐♥t❡r✈❛❧♦ I✱ ❝♦♥t❡♥❞♦ ♦ ♥ú♠❡r♦ r❡❛❧ l✱ ♣♦❞❡ s❡r t♦♠❛❞♦ ❞❛ ❢♦r♠❛ (l−
r, l+r)✱ ♦♥❞❡r é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ P♦rt❛♥t♦✱ ❞✐③❡r q✉❡ xn ❝♦♥✈❡r❣❡ ♣❛r❛ l✱ ✐st♦ é✱ q✉❡ lim
n→∞xn =l✱ é ♦ ♠❡s♠♦ q✉❡ ❞✐③❡r q✉❡ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧r >0✱ ❡①✐st❡
✷✳✷✳ ❆ ■❉❊■❆ ❉❊ ▲■▼■❚❊ ❉❊ ❯▼❆ ❙❊◗❯✃◆❈■❆
❖❜s❡r✈❡♠♦s ❛✐♥❞❛ q✉❡ ❛ ❝♦♥❞✐çã♦xn ∈(l−r, l+r)✱ ♣❛r❛ t♦❞♦ n > n0✱ ❡q✉✐✈❛❧❡
à |xn−l|< r ♣❛r❛ t♦❞♦n > n0✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✿
❆ ❞✐stâ♥❝✐❛ ❞❡xn ❛l s❡ t♦r♥❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛ ❞❡s❞❡ q✉❡n s❡❥❛ t♦♠❛❞♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳
➱ ✐♥t✉✐t✐✈♦ ♦ ❢❛t♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛(xn)♥ã♦ ♣♦❞❡r ❝♦♥✈❡r❣✐r ♣❛r❛ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s l1 ❡ l2 ❞✐st✐♥t♦s✱ ♣♦✐s✱ s❡ ❡st❡ ❢♦ss❡ ♦ ❝❛s♦✱ ♣♦❞❡rí❛♠♦s ❛❝❤❛r ❞♦✐s ✐♥t❡r✈❛❧♦s
❛❜❡rt♦s I1 ❡ I2 ❞✐s❥✉♥t♦s✱ ❝♦♥t❡♥❞♦ l1 ❡ l2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♣❛r❛
✈❛❧♦r❡s ❞❡ns✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✱ ♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ ❡st❛r✐❛♠ ❞❡♥tr♦ ❞❡ ❝❛❞❛
✉♠ ❞❡ss❡s ✐♥t❡r✈❛❧♦s✱ ♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧✳ ❆ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦ ❛♣❡♥❛s ❢♦r♠❛❧✐③❛ ❡st❛ ❛r❣✉♠❡♥t❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✾❪✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✶ ✭❯♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✮ ❙❡ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ l t❛❧ q✉❡
lim
n→∞xn =l✱ ❡♥tã♦ ❡❧❡ é ú♥✐❝♦✳
❊①❡♠♣❧♦ ✷✳✷✳✶ ▼♦str❡ q✉❡ limxn= 0✱ ♦♥❞❡xn =
1
n✳
❙♦❧✉çã♦
❉❛❞♦r > 0 ❛r❜✐trár✐♦✱ ♣♦❞❡♠♦s ♦❜t❡r n0 ∈N t❛❧ q✉❡ n0 >
1
r✳ ❊♥tã♦ n > n0 ⇒
1
n <
1
n0
< r✱ ♦✉ s❡❥❛✱ n > n0 ⇒
n1 −0
< r✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ nlim→∞
1
n = 0
❊①❡♠♣❧♦ ✷✳✷✳✷ ▼♦str❡ q✉❡ limxn= 0✱ ♦♥❞❡xn =
2n−1
n ✳
❙♦❧✉çã♦
Pr✐♠❡✐r♦ ✈❛♠♦s r❡❡s❝r❡✈❡r ❛ s❡q✉ê♥❝✐❛✳
xn=
2n−1
n =
2n
n −
1
n = 2−
1
n
✳
▲♦❣♦ ♦ ❧✐♠✐t❡ é 2✱ ♣♦✐s ❝❛❞❛ ✈❡③ q✉❡ ❛✉♠❡♥t❛♠♦s ♦ ✈❛❧♦r ❞❡ n ♦ r❡s✉❧t❛❞♦ ❞❡
2− 1
n ✜❝❛ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡ 2✳ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❢♦r♠❛❧ ♦❜s❡r✈❛♠♦s q✉❡
✷✳✷✳ ❆ ■❉❊■❆ ❉❊ ▲■▼■❚❊ ❉❊ ❯▼❆ ❙❊◗❯✃◆❈■❆
❞❛❞♦ r > 0 ❛r❜✐tár✐♦✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠ n0 ∈ N t❛❧ q✉❡ n0 >
1
r✳ ■ss♦ ♥♦s ❧❡✈❛ ❛
❝♦♥❝❧✉✐r q✉❡✿
n0 >
1
r ⇒
1
n0
< r⇒ − 1
n0
>−r ⇒2− 1
n0
>2−r
❈♦♠♦n > n0 ⇒
1
n <
1
n0
⇒ −1
n >−
1
n0
❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✿
2− 1
n >2−
1
n0
>2−r
✳
❙❛❜❡♥❞♦ q✉❡n > 0✱ t❡♠♦s q✉❡2− 1
n <2 ❡ 2−r <2−
1
n0
<2✱ ♣♦r ✜♠✱ r >0✱
❡♥tã♦ 2 < 2 +r ❡ 2−r < 2− 1
n0
< 2 +r✱ ❧♦❣♦ xn ∈ (2−r,2) ⊂ (2−r,2 +r)✱ ❡ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ |xn|<2 +r✱ ♦✉ s❡❥❛ |xn−2|< r✳
❊①❡♠♣❧♦ ✷✳✷✳✸ ▼♦str❡ q✉❡ limxn= 0✱ ♦♥❞❡xn =an ✱ ❝♦♠ 0< a <1✳ ❙♦❧✉çã♦ ❆✜r♠❛♠♦s q✉❡ lim
n→∞a
n = 0 ✭q✉❛♥❞♦ 0 < a < 1✮✳ ❈♦♠ ❡❢❡✐t♦✱ ❉❛❞♦
r > 0✱ ❝♦♠♦ 1
a >1✱ ❛s ♣♦tê♥❝✐❛s ❞❡ 1/a ❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡ ✐❧✐♠✐t❛❞❛
s✉♣❡r✐♦r♠❡♥t❡✳ ▲♦❣♦ ❡①✐st❡ n0 ∈N t❛❧ q✉❡ n > n0 ⇒
1
a
n
> 1
r✱ ♦✉ s❡❥❛ ✱
1
an >
1
r
✱ ✐st♦ é✱ an< r✳ ❆ss✐♠✱n > n
0 ⇒ |an−0|< r✱ ♦ q✉❡ ♠♦str❛ s❡r lim
n→∞a
n= 0✳
▲✐♠✐t❡s ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ♦♣❡r❛tór✐❛s q✉❡ t♦r♥❛♠ ♦ s❡✉ ❝á❧❝✉❧♦ ♠❛✐s ❢á✲ ❝✐❧✳ ◆❛ r❡❛❧✐❞❛❞❡✱ t❡r❡♠♦s ♣♦✉❝❛s ✈❡③❡s q✉❡ r❡❝♦rr❡r à ❞❡✜♥✐çã♦ ♣❛r❛ ❝❛❧❝✉❧❛r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❧✐♠✐t❡✱ ❜❛st❛♥❞♦ ♣❛r❛ ✐st♦ ✉t✐❧✐③❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ♦♣❡r❛tór✐❛s q✉❡ ❡st❛✲ ❜❡❧❡❝❡r❡♠♦s ❡ ❛❧❣✉♥s ♣♦✉❝♦s ❧✐♠✐t❡s ❢✉♥❞❛♠❡♥t❛✐s✱ ❡ss❡s✱ s✐♠✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦✳ ❖ ❢❛t♦ é q✉❡ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ♣❛r❛ ❞❡❞✉③✐r ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡r❛✐s✱ ❛✉♠❡♥t❛r❡♠♦s ❡♠ ♠✉✐t♦ ♦ ♥♦ss♦ ♣♦❞❡r ❞❡ ❝á❧❝✉❧♦✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✷ ✭▲✐♠✐t❡ ❞❛ s♦♠❛✮ ❙❡ lim
n→∞xn=l❡nlim→∞yn =k✱ ❡♥tã♦nlim→∞(xn+yn) =
lim
n→∞xn+ limn→∞yn =l+k ✳
✷✳✷✳ ❆ ■❉❊■❆ ❉❊ ▲■▼■❚❊ ❉❊ ❯▼❆ ❙❊◗❯✃◆❈■❆
❆ ♣r♦♣♦s✐çã♦ ✷✳✷✳✷ ❡st❛❜❡❧❡❝❡ ❛ ❛❞✐t✐✈✐❞❛❞❡ ❞♦s ❧✐♠✐t❡s✿ ♣❛r❛ s♦♠❛r ❞♦✐s ❧✐♠✐t❡s q✉❡ ❡①✐st❡♠✱ ♣♦❞❡♠♦s s♦♠❛r ❛s ❞✉❛s s❡q✉ê♥❝✐❛s ❡ ❝❛❧❝✉❧❛r ❛♣❡♥❛s ♦ ❧✐♠✐t❡ ❞❡st❛ s♦♠❛✳ ❊st❡ r❡s✉❧t❛❞♦ é ❜❛st❛♥t❡ ♥❛t✉r❛❧✱ ❜❛st❛♥❞♦✲♥♦s ♣❡♥s❛r ♦ s❡❣✉✐♥t❡✿ s❡xn♣♦❞❡ s❡r t♦r♥❛❞♦ tã♦ ♣ró①✐♠♦ ❞❡ l q✉❛♥t♦ q✉❡✐r❛♠♦s✱ ❞❡s❞❡ q✉❡ n s❡❥❛ s✉✜❝✐❡♥t❡♠❡♥t❡
❣r❛♥❞❡ ❡ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠yn✱ ❡♥tã♦ ❛ s♦♠❛xn+yn ♣♦❞❡ s❡r t♦r♥❛❞❛ tã♦ ♣ró①✐♠❛ ❞❡l+k q✉❛♥t♦ q✉❡✐r❛♠♦s t❛♠❜é♠✳ ❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛♥á❧♦❣❛ ✈❛❧❡ ♣❛r❛ ❛ ❞✐❢❡r❡♥ç❛
❡♥tr❡ ❧✐♠✐t❡s✳❆s ♣ró①✐♠❛s ♣r♦♣♦s✐çõ❡s ✭✷✳✷✳✸✱ ✷✳✷✳✹ ❡ ✷✳✷✳✺✮ ❡①♣❧♦r❛♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡ ♣❛r❛ ❛s ♦♣❡r❛çõ❡s ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ ❞✐✈✐sã♦✳ ❈❛s♦ ♦ ❧❡✐t♦r ❞❡s❡❥❡ ❛♥❛❧✐s❛r ❛s ❞❡♠❡♠♦♥str❛çõ❡s ❞❡ss❛s ♣r♦♣♦s✐çõ❡s ♣♦❞❡rá ❡♥❝♦♥trá✲❧❛s ❡♠ ❬✾❪✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✸ ✭▲✐♠✐t❡ ❞♦ ♣r♦❞✉t♦✮ ❙❡ lim
n→∞xn = l ❡ nlim→∞yn = k✱ ❡♥tã♦
lim
n→∞(xnyn) =
lim
n→∞xn nlim→∞yn
=lk ✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✹ ✭▲✐♠✐t❡ ❞❛ ♣♦tê♥❝✐❛✮ ❙❡ p ≥ 1 é ✉♠ ✐♥t❡✐r♦✱ ❡ lim
n→∞xn = l✱
❡♥tã♦ lim
n→∞(xn)
p
=lp✳
Pr♦♣♦s✐çã♦ ✷✳✷✳✺ ✭▲✐♠✐t❡ ❞♦ q✉♦❝✐❡♥t❡✮ ❙❡ lim
n→∞xn = l ❡ nlim→∞yn = k ✱ ❝♦♠
yn6= 0✱ ♣❛r❛ t♦❞♦ n ∈N✱ ❡ k6= 0✱ ❡♥tã♦ ❙❡ lim n→∞
xn
yn
= lim
n→∞xn
lim
n→∞yn
= l
k✳
❈❛♣ít✉❧♦ ✸
❙ér✐❡s
◆♦ ❡st✉❞♦ ❞❛s s♦♠❛s ❞❡ sér✐❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛ s♦♠❛ ❞♦s ✐♥✜♥✐t♦s t❡r♠♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❡s❝♦❧❛s ❜r❛s✐❧❡✐r❛s✱ ❧✐♠✐t❛✲s❡ tã♦ s♦♠❡♥t❡ ❛ ❛♥❛❧✐s❛r ♦s r❡s✉❧t❛❞♦s ❞❡ss❛s s♦♠❛s ❡♠ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s ❝♦♠ r❛③ã♦ ❡♥tr❡ −1 ❡ 1✳
❆❝r❡❞✐t❛✲s❡ q✉❡ ♦ ❛❧✉♥♦ s♦③✐♥❤♦ s❡❥❛ ❝❛♣❛③ ❞❡ ❝♦♥str✉✐r ♠ú❧t✐♣❧❛s r❡❧❛çõ❡s ❡♥✲ tr❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ s♦♠❛s ❝♦♠ ♣❛r❝❡❧❛s ✜♥✐t❛s ❡ ✐♥✜♥✐t❛s✳ ❈♦♥t✉❞♦✱ s❡ ♦s ❝♦♥❝❡✐t♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ❞❡ ❢♦r♠❛ ❢r❛❣♠❡♥t❛❞❛✱ ♠❡s♠♦ q✉❡ ❞❡ ❢♦r♠❛ ❝♦♠♣❧❡t❛ ❡ ❛♣r♦❢✉♥✲ ❞❛❞❛✱ ♥❛❞❛ ❣❛r❛♥t❡ q✉❡ ♦ ❛❧✉♥♦ ❡st❛❜❡❧❡ç❛ ❛❧❣✉♠❛ s✐❣♥✐✜❝❛çã♦ ♣❛r❛ ✐❞❡✐❛s ✐s♦❧❛❞❛s ❡ ❞❡s❝♦♥❡❝t❛❞❛s ✉♠❛s ❞❛s ♦✉tr❛s✳
❖ ♣r♦❢❡ss♦r ♣r❡❝✐s❛ ❛❧❡rt❛r s❡✉ ❛❧✉♥♦ ♣❛r❛ ♦ ❝✉✐❞❛❞♦ ❛♦ ♦♣❡r❛r ❝♦♠ s♦♠❛s ❞❡ ♣❛r✲ ❝❡❧❛s ✐♥✜♥✐t❛s✱ ♠♦str❛♥❞♦ q✉❡ ❡❧❛s t❛♥t♦ ♣♦❞❡♠ ❝♦♥✈❡r❣✐r ♣❛r❛ ✉♠ r❡s✉❧t❛❞♦ ❝♦♠♦ ♣♦❞❡♠ ❞✐✈❡r❣✐r✳ ◆♦ ❝❛s♦ ❞❡ ❝♦♥✈❡r❣✐r ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❝♦♠♦ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♦ q✉❡ ♥ã♦ é ♣♦ss✐✈❡❧ ❢❛③❡r s❡ ❛ sér✐❡ ❞✐✈❡r❣❡✳
✸✳✶✳ ❉❊❋■◆■➬Õ❊❙
✸✳✶ ❉❡✜♥✐çõ❡s
❉❡✜♥✐çã♦ ✸✳✶✳✶ ❉❛❞❛ ✉♠❛ s✉❝❡ssã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s (an)nǫN✱ ❝❤❛♠❛✲s❡ sér✐❡ ❞❡
♥ú♠❡r♦s r❡❛✐s ♦✉ sér✐❡ ♥✉♠ér✐❝❛ ❛ s♦♠❛ ✐♥✜♥✐t❛
a1+a2+a3+· · ·+an+· · ·=
+P∞
n=1
an
✳
❖❜s❡r✈❛çã♦ ✸✳✶✳✶ ❖s ♥ú♠❡r♦s a1, a2, . . . ❝❤❛♠❛♠✲s❡ t❡r♠♦s ❞❛ sér✐❡ ♥✉♠ér✐❝❛ ❡
♦ n✲és✐♠♦ t❡r♠♦ an é ❞❡s✐❣♥❛❞♦ ♣♦r t❡r♠♦ ❣❡r❛❧ ❞❛ sér✐❡✳
❊♠ r❡s✉♠♦✱ ✉♠❛ sér✐❡ é ✉♠❛ s♦♠❛s=a1+a2+· · ·+an+. . . ❝♦♠ ✉♠ ♥ú♠❡r♦ ✐♥✲ ✜♥✐t♦ ❞❡ ♣❛r❝❡❧❛s✳ P❛r❛ q✉❡ ✐ss♦ ❢❛ç❛ s❡♥t✐❞♦✱ ♣♦r❡♠♦ss = lim
n→∞(a1+a2 +· · ·+an)✳
❈♦♠♦ t♦❞♦ ❧✐♠✐t❡✱ ❡st❡ ♣♦❞❡ ❡①✐st✐r ♦✉ ♥ã♦✳ P♦r ✐ss♦ ❤á sér✐❡s ❝♦♥✈❡r❣❡♥t❡s ❡ sér✐❡s ❞✐✈❡r❣❡♥t❡s✳ ❆♣r❡♥❞❡r ❛ ❞✐st✐♥❣✉✐r ✉♠❛s ❞❛s ♦✉tr❛s é ❛ ♣r✐♥❝✐♣❛❧ ✜♥❛❧✐❞❛❞❡ ❞❡ss❛ s❡çã♦✳ ◆❛t✉r❛❧♠❡♥t❡✱ ♥ã♦ ♣♦❞❡♠♦s s♦♠❛r ✉♠ ❛ ✉♠✱ ♦s ✐♥✜♥✐t♦s t❡r♠♦s ❞❡ ✉♠❛ sér✐❡✱ ♦ q✉❡ ♣♦❞❡♠♦s ❢❛③❡r é s♦♠❛r ❝❛❞❛ ✈❡③ ♠❛✐s ♣❛r❝❡❧❛s✱ ❛✈❛❧✐❛♥❞♦ s❡ ❛♦ ❛❝r❡s❝❡♥t❛r ♠❛✐s t❡r♠♦s✱ ❛ s♦♠❛ ✈❛✐ s❡ ❛♣r♦①✐♠❛♥❞♦ ❞❡ ✉♠ ✈❛❧♦r r❡❛❧✳
✸✳✷ ❙♦♠❛s P❛r❝✐❛✐s
❈♦♥s✐❞❡r❡♠♦s ❛ sér✐❡a1+a2+a3+· · ·+an+· · ·=
+P∞
n=1
an ❡ ✈❛♠♦s ❢♦r♠❛r ✉♠❛ s❡q✉ê♥❝✐❛ {Sn} ❞❡ s♦♠❛s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
S1 =a1
S2 =a1+a2
S3 =a1+a2+a3
. . .
✸✳✸✳ ❙❖▼❆ ❉❖❙ ❚❊❘▼❖❙ ❉❊ ❯▼❆ ❙➱❘■❊ ●❊❖▼➱❚❘■❈❆
Sn=a1+a2+a3+· · ·+an =Sn−1+an= n X
k=1
ak
❆ s❡q✉ê♥❝✐❛ (Sn) é ❝❤❛♠❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ❞❛s s♦♠❛s ♣❛r❝✐❛✐s ❞❛ sér✐❡ ❡ Sn é ❝❤❛♠❛❞♦ ❞❡ n✲és✐♠❛ s♦♠❛ ♣❛r❝✐❛❧✳
❙❡ ❛ ❞✐❢❡r❡♥ç❛ |S −Sn|✱ ♦♥❞❡ S ∈ R✱ ♣✉❞❡r s❡r ❢❡✐t❛ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ ♣♦s✐t✐✈♦✱ ❞❡s❞❡ q✉❡ s❡ ❢❛ç❛ ♥ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞✐③❡♠♦s q✉❡ ❙ é ❛ s♦♠❛ ❞❛ sér✐❡✳ ❊♠ ❧✐♥❣✉❛❣❡♠ ♠❛✐s ❢♦r♠❛❧ t❡♠♦s q✉❡ ❞❛❞♦ r >0✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡
◆ t❛❧ q✉❡✱ ♣❛r❛n > N✱ é ✈❡r❞❛❞❡ q✉❡ |S−Sn|< r✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱S é ❛ s♦♠❛ ❞❛ sér✐❡ q✉❛♥❞♦ lim
n→∞(Sn) =S✳
❖❜s❡r✈❛çã♦ ✸✳✷✳✶ ◗✉❛♥❞♦ ✉♠❛ sér✐❡ ❛❞♠✐t❡ ✉♠❛ s♦♠❛ ❙✱ ❡ss❛ é ❝❧❛ss✐✜❝❛❞❛ ❝♦♠♦ ❝♦♥✈❡r❣❡♥t❡✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ lim
n→∞(Sn) ♥ã♦ ❡①✐st❡ ♦✉ é ✐♥✜♥✐t♦ ❛ sér✐❡ é ❞✐✈❡r✲
❣❡♥t❡✳
❊ss❡ ♣r♦❝❡❞✐♠❡♥t♦ é ✐♠♣♦rt❛♥tíss✐♠♦ ♣♦✐s ♥ã♦ s♦♠❛♠♦s ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣❛r✲ ❝❡❧❛s✳ ❆s s♦♠❛s Sn sã♦ ✜♥✐t❛s✱ ❥á q✉❡ ♥ é ✜♥✐t♦✳ ❊❧❛s sã♦ ✈❛❧♦r❡s ❛♣r♦①✐♠❛❞♦s ❞♦ q✉❡ ❝❤❛♠❛♠♦s ✏s♦♠❛✑ ❞❛ sér✐❡✳ ❖ q✉❡ ❛ ❞❡✜♥✐çã♦ ❞✐③ é q✉❡ ♦ ❡rr♦ q✉❡ s❡ ❝♦♠❡t❡ ❛♦ t♦♠❛r Sn ♥♦ ❧✉❣❛r ❞❡ S ♣♦❞❡ s❡r ❢❡✐t♦ tã♦ ♣❡q✉❡♥♦ q✉❛♥t♦ q✉❡r✐❛♠♦s✱ ❞❡s❞❡ ❞❡ q✉❡ ❢❛ç❛♠♦s ♥ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳
✸✳✸ ❙♦♠❛ ❞♦s t❡r♠♦s ❞❡ ✉♠❛ ❙ér✐❡ ❣❡♦♠étr✐❝❛
❆ sér✐❡a+ar+ar2
+ar3
+· · ·+arn−1
+. . . ♦✉
∞
P n=1
arn−1✱ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦ s❡
♦❜té♠ ❞♦ ♣r❡❝❡❞❡♥t❡ ♠✉❧t✐♣❧✐❝❛♥❞♦✲♦ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ r✭❛ r❛③ã♦✮ ❞❡s✐❣♥❛✲s❡ ♣♦r
sér✐❡ ❣❡♦♠étr✐❝❛✳
❙❡❥❛Sn ❛ ♥✲és✐♠❛ s♦♠❛ ♣❛r❝✐❛❧ t❡♠♦s✿
Sn=a+ar+ar
2
+ar3
+· · ·+arn−1 ✭✸✳✶✮
✸✳✹✳ ◆❖➬➹❖ ■◆❚❯■❚■❱❆ ❉❊ ❙➱❘■❊❙ ❈❖◆❱❊❘●❊◆❚❊❙
❈❛s♦ r ❂ ✶✱ t❡♠♦s q✉❡ Sn =na✳ ❈❛s♦ r6= 1✱ ♠✉❧t✐♣❧✐❝❛♥❞♦Sn ♣♦r r t❡♠♦s✿
rSn=ar+ar
2
+ar3
+ +ar4
+· · ·+arn−1
+arn ✭✸✳✷✮
s✉❜tr❛✐♥❞♦ ✸✳✶ ❞❡ ✸✳✷✱ ❡♥❝♦♥tr❛♠♦s✿
rSn−Sn=arn−a⇒Sn(r−1) = a(rn−1)⇒Sn=
a(rn−1)
(r−1)
❙❡0< r <1✱ t❡♠♦s q✉❡ lim
x→∞r
n = 0 ✭❡①❡♠♣❧♦ ✷✳✷✳✸✮✱ ❧♦❣♦ ❝♦♥❝❧✉í♠♦s q✉❡✿
∞
X
n=1
arn−1
= lim
x→∞
a(rn−1)
(r−1) =
a
1−r
✸✳✹ ◆♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ❙ér✐❡s ❝♦♥✈❡r❣❡♥t❡s
❯♠ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦
❚♦♠❡♠♦s ❛ s✉❝❡ssã♦ ❣❡♦♠étr✐❝❛ ❞❡ t❡r♠♦ ❣❡r❛❧
a1 =
1 2 ❡r =
1 2 ✿ N P n=1 1 2n =
1 2+
1 22 +
1 23 +
1
24 +· · ·+
1 2N =
1 2×
1− 1
2N
1− 1
2
= 1− 1
2N ❖ q✉❡ ❛❝♦♥t❡❝❡ ❛ ❡st❛ ✐❣✉❛❧❞❛❞❡ s❡ t♦♠❛r♠♦s ♦ ❧✐♠✐t❡ N →+∞❄
❚❡♠✲s❡✿
lim
N→+∞SN = limN→+∞
N P n=1
1
2n = limx→+∞1−
1 2N = 1
■st♦ s✐❣♥✐✜❝❛ q✉❡✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ s❡ s♦♠❛r♠♦s ❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣❛r❝❡❧❛s✳
1 2+
1 4+
1
8+· · ·+ 1 2100 +
1
2101 +· · ·+
1 21000 +
1
21001 +. . .
♥ã♦ ♦❜t❡♠♦s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❝♦♠♦ ♣♦❞❡rí❛♠♦s ♣❡♥s❛r✳ ❖❜t❡♠♦s s✐♠✲ ♣❧❡s♠❡♥t❡ ♦ ✈❛❧♦r 1✳ ❊s❝r❡✈❡♠♦s✿
+P∞
✸✳✺✳ ❙➱❘■❊ ❍❆❘▼Ô◆■❈❆
❊st❡ r❡s✉❧t❛❞♦ ♥ã♦ é ❛ss✐♠ tã♦ ❡s♣❛♥t♦s♦✳ ➱ ✈❡r❞❛❞❡ q✉❡ ❡st❛♠♦s ♥✉♠ ❝❡rt♦ s❡♥t✐❞♦ ❛ s♦♠❛r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣❛r❝❡❧❛s ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛s✳ ▼❛s t❛♠❜é♠ é ✈❡r❞❛❞❡ q✉❡ ❡ss❛s ♣❛r❝❡❧❛s sã♦ ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s✳ ❆❧✐ás✱ ❡st❡ r❡s✉❧t❛❞♦ é ♠✉✐t♦ ❢á❝✐❧ ❞❡ ♣❡r❝❡❜❡r ✐♥t✉✐t✐✈❛♠❡♥t❡✿ ♦❜té♠✲s❡ ✉♠❛ ❜♦❛ ✐❧✉str❛çã♦ ❞❡st❛ ✐❣✉❛❧❞❛❞❡ t♦♠❛♥❞♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1 ❡ ❞✐✈✐❞✐♥❞♦✲♦ s✉❝❡ss✐✈❛♠❡♥t❡ ❛♦ ♠❡✐♦✳
❋✐❣✉r❛ ✸✳✶✿ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛
✸✳✺ ❙ér✐❡ ❍❛r♠ô♥✐❝❛
❖ r❛❝✐♦❝í♥✐♦ ✐♥t✉✐t✐✈♦ ✈✐st♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r ♥❡♠ s❡♠♣r❡ ❞á ❝❡rt♦ ✭❝♦♠♦ ♦❝♦rr❡ ❝♦♠ ❣r❛♥❞❡ ♣❛rt❡ ❞♦s r❛❝✐♦❝í♥✐♦s ❞❡st❡ t✐♣♦ ❡♠ ♠❛t❡♠át✐❝❛✮✳ ❱❛♠♦s ❛♥❛❧✐s❛r ♦ ❝❛s♦ ❞❛ ❝❤❛♠❛❞❛ sér✐❡ ❤❛r♠ô♥✐❝❛ ❛❜❛✐①♦✿
∞
X
n=1
1
n = 1 +
1 2+
1 3+
1
4+· · ·+ 1
n +. . .
❈♦♠♦ ❛s ♣❛r❝❡❧❛s ✭♣♦s✐t✐✈❛s✮ ❛ s❡r❡♠ s♦♠❛❞❛s sã♦ ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s ❡ t❡♥❞❡♥❞♦ ❛ 0✱ ❛ ✐♥t✉✐çã♦ ♥♦s ❧❡✈❛r✐❛ ❛ ❛❝r❡❞✐t❛r q✉❡ t❛❧ sér✐❡ s❡r✐❛ ❝♦♥✈❡r❣❡♥t❡✱ ♦ q✉❡ ♥ã♦ é
✈❡r❞❛❞❡✳ ❈♦♥s✐❞❡r❡ ❛♣❡♥❛s ❛s s♦♠❛s ♣❛r❝✐❛✐s✿ S2n =
2n
P
k=1
1
k = 1 +
1 2 + 1 3 + 1 4 + 1 5+ 1
6+. . . 1
2n−1+. . .
1
2n✳ ❊ ♥♦t❡ q✉❡✿ 1 3 + 1 4 > 1 4+ 1 4 = 1 2❀ 1 5+ 1 6+ 1 7+ 1 8 > 1 8 + 1 8 + 1 8 + 1 8 = 1 2❀ . . . ❀
1 2n−1 + 1 +
1
2n−1+ 2 +· · ·+
1 2n
> 1
2n +· · ·+
1 2n =
1
2✱ ❆ss✐♠✿ S2n >1 + 1 2+
1 2+
1
2+· · ·+ 1
2 = 1 +n
1 2
❡ ❝♦♠♦ 1 +n
2 ♥ã♦ é ❧✐♠✐t❛❞❛
s✉♣❡r✐♦r♠❡♥t❡✱ ❡♥tã♦ S2n t❛♠❜é♠ ♥ã♦ é ❡✱ ♣♦rt❛♥t♦✱ Sn ♥ã♦ é ❧✐♠✐t❛❞❛ ❡✱ s❡♥❞♦
❛ss✐♠✱ ❞✐✈❡r❣❡✳
✸✳✻✳ ❍■❙❚Ó❘■❆ ❉❖ ❳❆❉❘❊❩
❈♦♠♦ ❝❛❞❛ ♣❛r❝❡❧❛ ❡♥tr❡ ♣❛rê♥t❡s❡s é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 1
2✱ t❡♠♦s q✉❡ ❛ s♦♠❛ ❞❡
t♦❞❛s ❛s ♣❛r❝❡❧❛s ♣♦❞❡ s❡r ♠✐♥♦r❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♣❛r❝❡❧❛s ✐❣✉❛✐s ❛ 1
2✱ q✉❡
t❡♠ s♦♠❛ ✐♥✜♥✐t❛✳ P♦r ✐ss♦ é ✐♠♣♦rt❛♥t❡ ❝♦♥❤❡❝❡r ❢♦r♠❛❧♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ sér✐❡s ❝♦♥✈❡r❣❡♥t❡s ♣❛r❛ ❡♥t❡♥❞❡r ♦ q✉❡ ♦❝♦rr❡ ❡♠ ❝❛s♦s ❝♦♠♦ ❡st❡✳
✸✳✻ ❍✐stór✐❛ ❞♦ ①❛❞r❡③
❙❡❣✉♥❞♦ ❛ ❧❡♥❞❛ q✉❡ ❥✉st✐✜❝❛ ❛ ♦r✐❣❡♠ ❞❛ ❝r✐❛çã♦ ❞♦ ①❛❞r❡③✱ ♦ ❥♦❣♦ t❡r✐❛ s✐❞♦ ❝r✐❛❞♦ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝✉r❛r ❛ ❞❡♣r❡ssã♦ ❞❡ ✉♠ r❡✐✳ ❊ss❡ t❡r✐❛ ✜❝❛❞♦ tã♦ ❡♥❝❛♥t❛❞♦ ❝♦♠ ♦ ❥♦❣♦ q✉❡ ♦❢❡r❡❝❡✉ ❛♦ ✐♥✈❡♥t♦r ❛ r❡❝♦♠♣❡♥s❛ q✉❡ ❡❧❡ q✉✐s❡ss❡✳ ❖ ✐♥✈❡♥t♦r ❛♣❛r❡♥t❡♠❡♥t❡ ♣❡❞✐✉ ♣♦✉❝♦✳ ❆♣❡♥❛s ✉♠ t❛❜✉❧❡✐r♦ ❝❤❡✐♦ ❞❡ tr✐❣♦✱ ♠❛s ❞❡ ♠♦❞♦ q✉❡ ♥❛ ♣r✐♠❡✐r❛ ❝❛s❛ ❤♦✉✈❡ss❡ ✉♠ ❣rã♦✱ ♥❛ s❡❣✉♥❞❛✱ ❞♦✐s✱ ♥❛ t❡r❝❡✐r❛✱ q✉❛tr♦✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❞♦❜r❛♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣rã♦s ❛té ❛ ❝❛s❛ ✻✹✳ ▲❡♥❞❛ ♦✉ ♥ã♦✱ ❡ss❛ ❤✐stór✐❛ é ✉♠ ❡①❝❡❧❡♥t❡ r❡❝✉rs♦ ❞✐❞át✐❝♦✿ ❛❧é♠ ❞❡ ❡✈✐❞❡♥❝✐❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ é ✉♠ ❡①❝❡❧❡♥t❡ ❡①❡r❝í❝✐♦ ❞❡ s♦♠❛tór✐♦ ❞❡ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s✳
1 + 2 + 22
+ 23
+ 24
+· · ·+ 263
❆♦ s❡r s✉❣❡r✐❞♦ ❛ ✉♠ ❛❧✉♥♦ q✉❡ ❝❛❧❝✉❧❛ss❡ ♦ ❞é❜✐t♦ ❞♦ r❡✐✱ ❡ss❡ ❝❤❡❣❛r✐❛ ❛♦ r❡s✉❧t❛❞♦ 264
−1✱ ♦ q✉❡ s❡r✐❛ q✉❛s❡ ✶✽ ✏q✉✐♥t✐❧❤õ❡s✑ ❞❡ ❣rã♦s ✭♦ ♥ú♠❡r♦ ✶ s❡❣✉✐❞♦
❞❡ ✷✵ ③❡r♦s✮✳ ❯♠❛ q✉❛♥t✐❞❛❞❡ ❡s♣❛♥t♦s❛♠❡♥t❡ ❛❧t❛✳
❊♠ ❬✹❪ é s✉❣❡r✐❞♦ ✉♠❛ r❡s♣♦st❛ ❛ t❛❧ ♣❡❞✐❞♦✿ s❡r✐❛ ♦❢❡r❡❝✐❞♦ ❛♦ ✐♥✈❡♥t♦r ❞♦ ❥♦❣♦ ✉♠❛ ♦❢❡rt❛ ❛✐♥❞❛ ♠❛✐s ❣❡♥❡r♦s❛✳ ◆♦ ❧✉❣❛r ❞❡ ✉♠ t❛❜✉❧❡✐r♦ ❝♦♠ 64 ❝❛s❛s✱ s❡r✐❛
♦❢❡r❡❝✐❞♦ ✉♠ t❛❜✉❧❡✐r♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❝❛s❛s✳ ❆ss✐♠ ❛ ♥♦✈❛ ❞í✈✐❞❛ s❡r✐❛✿
S = 1 + 2 + 22
+ 23
+ 24
+· · ·+ 263
+ 264
+ 265
+. . .
❊ss❡ ♣❛❣❛♠❡♥t♦ s❡r✐❛ ❢❡✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❖ q✉❡ ❧❤❡ ❤❛✈✐❛ s✐❞♦ ♣r♦♣♦st♦ s❡r✐❛
✸✳✻✳ ❍■❙❚Ó❘■❆ ❉❖ ❳❆❉❘❊❩
♣❛❣♦ ❡♠ ❞♦❜r♦✱ ♦✉ s❡❥❛✱
2s= 2 1 + 2 + 22
+ 23
+ 24
+· · ·+ 263
+ 264
+ 265
+. . .
❉❡s❞❡ q✉❡ ♦ ✐♥✈❡♥t♦r ❞❡ss❡ ❝♦♠♦ tr♦❝♦ ♦ q✉❡ ❤❛✈✐❛ s✐❞♦ ❝♦♥❜✐♥❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡✳ ■♥t✉✐t✐✈❛♠❡♥t❡ ❡ss❛ ♦♣❡r❛çã♦ ♥ã♦ t❡r✐❛ ♣r♦❜❧❡♠❛ ✉♠❛ ✈❡③ q✉❡ ♦ ✈❛❧♦r ✭D✮ ❞❛ ❞í✈✐❞❛
r❡❛❧ ♣❛r❛ ♦ ✐♥✈❡♥t♦r s❡r✐❛ D = 2S −S✳ ❈✉r✐♦s❛♠❡♥t❡✱ ❛♦ s❡r ❢❡✐t♦ ♥♦✈❛♠❡♥t❡ ♦
❝á❧❝✉❧♦ ❞❛ ❞í✈✐❞❛ ❞♦ r❡✐ ✿
2s−s = 2 + 22
+ 23
+ 24
+· · ·+ 263
+ 264
+ 265
+. . .
− 1 + 2 + 22
+ 23
+ 24
+· · ·+ 263
+ 264
+ 265
+. . .
= −1 + (2−2) + 22
−22
+ 23
−23
+ 24
−24
+. . .
+ 263
−263
+ 264
−264
+ 265
−265
+. . .
⇒ D=−1
t♦❞❛s ❛s ❞❡♠❛✐s ♣❛r❝❡❧❛s ✭✐♥✜♥✐t❛s✮ sã♦ ❝❛♥❝❡❧❛❞❛s✳ ◆♦ ✜♥❛❧ ❞❛s ❝♦♥t❛s✱ ♦ ♣♦❜r❡ ❝r✐❛❞♦r ❞♦ ①❛❞r❡③ ❛❝❛❜♦✉ ❛✐♥❞❛ ❞❡✈❡♥❞♦ ✉♠ ❣rã♦(−1)❞❡ tr✐❣♦ ❛♦ r❡✐✳ ➱ ❝❧❛r♦ q✉❡ t❛❧
❛rt✐❢í❝✐♦ ♥ã♦ ❢❛③ ❥✉st✐ç❛ à s❛❣❛❝✐❞❛❞❡ ♦r✐❣✐♥❛❧ ❞❛ ❤✐stór✐❛✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ss❛ s♦❧✉çã♦ é ❡st❡♥❞❡r✱ s❡♠ ♠❛✐♦r❡s ❝✉✐❞❛❞♦s✱ ♣❛r❛ s♦♠❛s ✐♥✜♥✐t❛s✱ ♣r♦❝❡ss♦s s❛❜✐❞❛♠❡♥t❡ ✈á❧✐❞♦s ♣❛r❛ s♦♠❛s ✜♥✐t❛s✳
❯♠ ❢❛t♦ ❝✉r✐♦s♦ é q✉❡ ❡ss❡ ♣r♦❜❧❡♠❛ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❛ ♠✐♥❤❛ t✉r♠❛ ❞❡ ♠❡s✲ tr❛❞♦ ✭Pr♦❢♠❛t✮ ❡ ♥❡♥❤✉♠ ❞♦s ♠❡✉s ❝♦❧❡❣❛s ♣❡r❝❡❜❡✉ ❛❧❣✉♠❛ ❝♦✐s❛ ❡rr❛❞❛ ❝♦♠ ❛s ♦♣❡r❛çõ❡s r❡❛❧✐③❛❞❛s ❛♦ ❧♦♥❣♦ ❞❡ss❡ ♣r♦❜❧❡♠❛✳
❈❛♣ít✉❧♦ ✹
▼ét♦❞♦ ❞❡ ❊①❛✉stã♦
✹✳✶ ❊✉❞ó①✐♦ ❡ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦
❖ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♣♦r Pr✐♥❝í♣✐♦ ❞❡ ❊✉❞ó①✐♦✲❆rq✉✐♠❡❞❡s✱ ♣♦r t❡r ♥❛ s✉❛ ❜❛s❡ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❛♣r❡s❡♥t❛❞❛ ♣♦r ❊✉❞ó①✐♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽✲ ✸✺✺ ❛✳ ❈✳✮ ❡ ♣♦r ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼✲✷✶✷ ❛✳❈✳✮ t❡r s✐❞♦ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐♦r ✈✐s✐❜✐❧✐❞❛❞❡ ❧❤❡ ❞❡✉✳❬✶✷❪
❊✉❞♦①♦ ❛♣r❡s❡♥t♦✉ ❛ s✉❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❝♦♠♦ ♠♦❞♦ ❞❡ ✉❧tr❛♣❛ss❛r ❛s ❧✐♠✐t❛çõ❡s ♥❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛ ❡✈✐❞❡♥❝✐❛❞❛s ❝♦♠ ❛ ❞❡s❝♦❜❡rt❛ ❞♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s✱ q✉❡ ❞❡✐t❛✈❛ ♣♦r t❡rr❛ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❞♦s ♣✐t❛❣ór✐❝♦s✳ ❆rq✉✐♠❡❞❡s ❛♣❧✐❝♦✉ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ ♣❛r❛ ♣r♦✈❛r ✐♥ú♠❡r♦s r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛ ❝♦♠♣r✐♠❡♥t♦s✱ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❞✐✈❡rs❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ t❛♠❜é♠ ❛♦ ❝á❧❝✉❧♦ ❞❡ ❝❡♥tr♦s ❞❡ ❣r❛✈✐❞❛❞❡✳
❖ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ é ♦ ❢✉♥❞❛♠❡♥t♦ ❞❡ ✉♠ ❞♦s ♣r♦❝❡ss♦s ❡ss❡♥❝✐❛✐s ❞♦ ❝á❧❝✉❧♦ ✐♥✜♥✐t❡s✐♠❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❡♥q✉❛♥t♦ ♥♦ ❝á❧❝✉❧♦ s❡ s♦♠❛ ✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ♣❛r❝❡✲ ❧❛s✱ ❆rq✉✐♠❡❞❡s ♥✉♥❝❛ ❝♦♥s✐❞❡r♦✉ q✉❡ ❛s s♦♠❛s t✐✈❡ss❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ t❡r♠♦s✳ P❛r❛ ♣♦❞❡r ❞❡✜♥✐r ✉♠❛ s♦♠❛ ❞❡ ✉♠❛ sér✐❡ ✐♥✜♥✐t❛ s❡r✐❛ ♥❡❝❡ssár✐♦ ❞❡s❡♥✈♦❧✈❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ r❡❛❧ q✉❡ ♦s ❣r❡❣♦s ♥ã♦ ♣♦ss✉í❛♠✳ ◆ã♦ é✱ ♣♦✐s✱ ❝♦rr❡t♦ ❢❛❧❛r
✹✳✶✳ ❊❯❉Ó❳■❖ ❊ ❖ ▼➱❚❖❉❖ ❉❊ ❊❳❆❯❙❚➹❖
❞♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❣❡♦♠étr✐❝♦ ❞❡ ♣❛ss❛❣❡♠ ♣❛r❛ ♦ ❧✐♠✐t❡✳ ❆ ♥♦çã♦ ❞❡ ❧✐♠✐t❡ ♣r❡ss✉♣õ❡ ❛ ❝♦♥s✐❞❡r❛çã♦ ❞♦ ✐♥✜♥✐t♦ q✉❡ ❡st❡✈❡ s❡♠♣r❡ ❡①❝❧✉í❞❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛✱ ♠❡s♠♦ ❡♠ ❆rq✉✐♠❡❞❡s✳ ▼❛s✱ ♥♦ ❡♥t❛♥t♦✱ ♦ s❡✉ tr❛❜❛❧❤♦ ❢♦✐✱ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ♦ ♠❛✐s ❢♦rt❡ ✐♥❝❡♥t✐✈♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣♦st❡r✐♦r ❞❛s ✐❞❡✐❛s ❞❡ ❧✐♠✐t❡ ❡ ❞❡ ✐♥✜♥✐t♦ ♥♦ sé❝✉❧♦ ❳■❳✳ ❉❡ ❢❛t♦✱ ♦s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s ❝♦♥st✐t✉ír❛♠ ❛ ♣r✐♥❝✐♣❛❧ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛ ❞♦ sé❝✉❧♦ ❳❱■■ q✉❡ ❞❡s❡♠♣❡♥❤♦✉ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ✐♥✜♥✐t❡s✐♠❛❧✳
❉❡✜♥✐çã♦ ✹✳✶✳✶ ❙❡❥❛♠ ❞❛❞❛s q✉❛tr♦ ❣r❛♥❞❡③❛s a✱ b✱ c ❡ d ❡ s✉❛s r❛③õ❡s a b ❡
c d✳
❚❡♠♦s q✉❡ a
b = c
d s❡✱ ♣❛r❛ t♦❞❛ ❢r❛çã♦ m
n✱ ❛❝♦♥t❡❝❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❝❛s♦s✿
• ❖✉ m
n < a b ❡ m n < c
d✱ ✐st♦ é✱ ❛ ❢r❛çã♦ é ♠❡♥♦r q✉❡ ❛♠❜❛s❀
• ❖✉ m
n = a b ❡ m n = c
d ✐st♦ é✱ ❛ ❢r❛çã♦ é ✐❣✉❛❧ ❛ ❛♠❜❛s❀
• ❖✉ m
n > a b ❡ m n > c
d✐st♦ é✱ ❛ ❢r❛çã♦ é ♠❛✐♦r q✉❡ ❛♠❜❛s✳
❖✉ s❡❥❛✱ s❡ a
b = c
d ♥ã♦ ♣♦❞❡♠♦s t❡r ✉♠❛ ❢r❛çã♦ q✉❡ ❡st❡❥❛ ❡♥tr❡ a b ❡
c d✳
❯s❛♥❞♦ ❡st❛ ✐❞❡✐❛ ❡ ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s ♥ã♦ é ❧✐♠✐t❛❞♦ s✉♣❡r✐✲ ♦r♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❞♦✐s r❡s✉❧t❛❞♦s✿
❚❡♦r❡♠❛ ✹✳✶✳✶ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ a > 0 ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n0 > 0 t❛❧ q✉❡
1
n0
< a✳
❉❡♠♦♥str❛çã♦✿ ❆ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ é s✐♠♣❧❡s✱ ♣♦✐s ❞❛❞♦n ∈N❡①✐st❡♠ três
♦♣çõ❡s ♣❛r❛ ✉♠❛ ❢r❛çã♦ 1
n✳ P♦❞❡♠♦s t❡r
1
n < a✱ ❡ ❡♥tã♦ ♥❛❞❛ ❤á ❛ ♣r♦✈❛r✳ P♦❞❡♠♦s t❡r 1
n = a ❡ ❛ss✐♠
1
n+1 < a✳ ❙✉♣♦♥❤❛♠♦s ❡♥tã♦✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ❡st❡s ❞♦✐s ❝❛s♦s
♥ã♦ ♣♦ss❛♠ ❛❝♦♥t❡❝❡r✳ ❊♥tã♦ 1
n > a ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✳ ❚❡r❡♠♦s q✉❡n < 1
a❀ ∀n∈N♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✱ ♦ q✉❡
é ✉♠ ❛❜s✉r❞♦✳ ▲♦❣♦✱ ❡①✐st❡ n0 t❛❧ q✉❡
1
n0
< a✳