❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ◆●❯❨➍◆ ❚❍➚ ❍➪❆ ▲Þ ❚❍❯❨➌❚ ❈❍❯➱■ P❍❹◆ ❑Ý ❱⑨ Ù◆● ệ P ữớ ữợ ❚❙✳ ◆●❯❨➍◆ ❱❿◆ ❍⑨❖ ❍⑨ ◆❐■ ✲ ✐ 2014 ▲❮■ ữủ ỷ ỡ tợ ❝→❝ ●✐↔♥❣ ✈✐➯♥ ❦❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❍➔♦ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ❝❤➳✳ ❊♠ ỡ ữủ ỳ ỵ õ õ qỵ t ổ ✤➸ ❜↔♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ♥❤÷ ❤✐➺♥ t↕✐✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❍á❛ ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❍➔♦ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ợ t ỵ tt ộ ý ự ữủ t ổ trũ ợ t t➔✐ ♥➔♦ ❦❤→❝✳ ◆➳✉ s❛✐ ❡♠ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ❱✐➯♥ ◆❣✉②➵♥ ❚❤à ❍á❛ ✐✐✐ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ◆ë✐ ❞✉♥❣ ✶ ✺ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ✻ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶✳✶✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷✳ ✣✐➲✉ ❦✐➺♥ ✤➸ ❝❤✉é✐ ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸✳ ❚➼♥❤ ❝❤➜t ✈➲ ❝→❝ ♣❤➨♣ t♦→♥ ❝õ❛ ❝❤✉é✐ ❤ë✐ tö ✳ ✶✳✶✳✹✳ ❉➜✉ ❤✐➺✉ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❞÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✺✳ ❉➜✉ ❤✐➺✉ s♦ s→♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✻✳ ❉➜✉ ❤✐➺✉ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✼✳ ❉➜✉ ❤✐➺✉ ❉✬❆❧❛♠❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✽✳ ❉➜✉ ❤✐➺✉ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✾✳ ❈❤✉é✐ ✈ỵ✐ sè ❤↕♥❣ ❝â ❞➜✉ tũ ỵ ✶✳✷✳ ❈❤✉é✐ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤✉é✐ sè ✐✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✽ ✶✵ ✶✵ ✶✶ ✶✸ ✶✹ ✶✺ ✶✻ ✷✵ ✷✵ ✶✳✷✳✷✳ ❙ü ❤ë✐ tö ✈➔ ❤ë✐ tö ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸✳ ❈→❝ t✐➯✉ ❝❤✉➞♥ ❤ë✐ tö ✤➲✉ ❝õ❛ ❝❤✉é✐ ❤➔♠ sè ✳ ✳ ✳ ✶✳✷✳✹✳ ❚➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❤➔♠ sè ❤ë✐ tö ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳ ❈❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶✳ ❑❤→✐ ♥✐➺♠ ✈➲ ❝❤✉é✐ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷✳ ❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸✳ ❑❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ ❧ơ② t❤ø❛ ❝õ❛ ♠ët sè ❤➔♠ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❱➜♥ ✤➲ C1 −❦❤↔ tê♥❣ ✈➔ →♣ ❞ư♥❣ ❝õ❛ C1 −❦❤↔ ✸✵ tê♥❣ ✤è✐ ✈ỵ✐ ❝❤✉é✐ ❋♦✉r✐❡r ✷✳✶✳ ✷✳✷✳ ✷✳✸✳ ✷✳✹✳ ✷✶ ✷✶ ✷✻ ✷✼ ✷✼ ✷✽ ✸✸ ❑❤→✐ ♥✐➺♠ ✈➲ C1−q✉→ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ✈➼ ❞ö ✈➲ C1−❦❤↔ tê♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ✤✐➲✉ ❦✐➺♥ ✤➸ ❝❤✉é✐ C1−❦❤↔ tê♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ⑩♣ ❞ư♥❣ ❝õ❛ C1−tê♥❣ ✤è✐ ✈ỵ✐ ❝❤✉é✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✷✳✹✳✶✳ ❈❤✉é✐ ❧÷đ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷✳ ❈❤✉é✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✸✳ ⑩♣ ❞ư♥❣ ❝õ❛ C1−tê♥❣ ✤è✐ ✈ỵ✐ ❝❤✉é✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✹ ✸✻ ✹✵ ✹✵ ✹✶ ✹✶ ❑➳t ❧✉➟♥ ✹✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✾ ✈ ▼Ð ỵ t ỵ tt ộ số ụ ữ ỵ tt ộ ữủ ự tứ rt sợ ữ t ❤♦➔♥ t❤✐➺♥ ♠ët ❝→❝❤ ❝❤✉➞♥ ♠ü❝✳ ❈→❝ ❦➳t q✉↔ ✤➭♣ ♥❤➜t tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥➔② ♣❤↔✐ ♥â✐ ✤➳♥ ❝→❝ ❝æ♥❣ tr➻♥❤ t➼♥❤ t♦→♥ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝ ▲✳ ❊✉❧❡r ❝ò♥❣ ♠ët sè ♥❤➔ ❚♦→♥ ❤å❝ ✤÷ì♥❣ t❤í✐✳ ❚✉② ♥❤✐➯♥ ❝→❝ t q trữợ õ ỹ ✤➳♥ ♠ët sè ♥❤➔ ❚♦→♥ ❤å❝ ♥❤÷ ▲❡✐❜♥✐③✱ ◆❡✇t♦♥ ✈➔ sỹ ỵ tt ộ ữủ t❤➔♥❤ ♠ët ❝→❝❤ ❦❤→ tü ♥❤✐➯♥ ①✉➜t ♣❤→t tø ❝→❝ ❝ỉ♥❣ tr➻♥❤ t➼♥❤ t♦→♥ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ t❤í✐ ✤â tø ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ t❤ü❝ t➳✳ ❱➲ ❧➽♥❤ ✈ü❝ ♥➔②✱ t❤❡♦ t✐➳♥ tr➻♥❤ ❧à❝❤ sû ❝â ❧➩ ♣❤↔✐ ❦➸ ✤➳♥ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➲ ❝❤✉é✐ ❤➻♥❤ ❤å❝ {1 + x + x2 + x3 + · · · } ✳ ❈❤✉é✐ ❤➻♥❤ ❤å❝ ①✉➜t ❤✐➺♥ ♥❤÷ ♠ët ❦➳t q✉↔ ❦❤ỉ♥❣ ❦➳t t❤ó❝ tr♦♥❣ ♣❤➨♣ ❝❤✐❛ −1 x ✳ ❱➜♥ ✤➲ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ t❤❡♦ ♥❣❤➽❛ ❤✐➺♥ ✤↕✐ ✤➣ ✤÷đ❝ ①✉➜t ❤✐➺♥ tø rt sợ tr ỵ ✣✐➲✉ ✤â ❦❤æ♥❣ ♥❣↕❝ ♥❤✐➯♥ ❦❤✐ ♥❤➔ ❚♦→♥ ❤å❝ ▲✳ ❊✉❧❡r sû ❞ö♥❣ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❝❤✉é✐ ❤➻♥❤ ❤å❝ + x + x2 + x3 + · · · = 1−x ✤➸ ❦❤➥♥❣ ✤à♥❤ q✉↔ q✉②➳t r➡♥❣ 1 − + − + = ; ✶ ✈ỵ✐ x = −1 ✈➔ 1 − + 2 − 23 + · · ã = ; ợ x = ữỡ tỹ ♥❤÷ t❤➳✱ tø ❜✐➸✉ ❞✐➵♥ 1−x = + 2x + 3x2 + 4x3 + ▲✳❊✉❧❡r ❝ô♥❣ ❦❤➥♥❣ ✤à♥❤ ✤➥♥❣ t❤ù❝ s❛✉ 1 − + − + = ; ✈ỵ✐ x = −1 ✣✐➲✉ t❤ü❝ t➳ r➡♥❣✱ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✤÷ì♥❣ t❤í✐ ❝ơ♥❣ r➜t ♥❣❤✐ ♥❣í ♥❤ú♥❣ ❦➳t q✉↔ ✤÷❛ r❛ tr➯♥ ✤➙②✳ ❚✉② ♥❤✐➯♥✱ ❤å ❝ơ♥❣ ❦❤ỉ♥❣ ❝â sü ❤✐➸✉ ❜✐➳t ♠ët ❝→❝❤ t❤➜✉ ✤→♦ ✤➸ ♣❤õ ✤à♥❤ ❤❛② ❝❤➜♣ ♥❤➟♥ ♥❤ú♥❣ ❦➳t q✉↔ ♥❤÷ t❤➳✳ ❘ã r➔♥❣ ❝→❝ ❦❤➥♥❣ ✤à♥❤ ð tr➯♥ ✈➲ ❝ì ❜↔♥ ❦❤ỉ♥❣ ❝❤➼♥❤ ①→❝✳ ✣➸ t❤➜② ✤÷đ❝ ✤✐➲✉ ✤â t❛ ❝â t❤➸ ①➨t ❝❤✉é✐ − + − + t❤❡♦ ❝→❝❤ s✉② ❧✉➟♥ tr➯♥ ✤➙② ❝â ❦➳t q✉↔ ❜➡♥❣ 12 ✳ ❚❤➳ ♥❤÷♥❣✱ tø ❜✐➸✉ ❞✐➵♥ 1+x − x2 = = − x2 + x3 − x5 + x6 − x8 + 1+x+x 1x ợ x=1 t ữủ q − + − + = ❈❤➾ ✤ì♥ ❣✐↔♥ ♥❤÷ t❤➳ ❝ơ♥❣ ✤➣ ❝❤➾ r❛ r➡♥❣ ♥❤ú♥❣ s✉② ❧✉➟♥ ❝õ❛ ❊✉❧❡r ❦❤æ♥❣ ❝❤➼♥❤ ①→❝✳ ❈❛✉❝❤② ✈➔ ❆❜❡❧ ❧➔ ♥❤ú♥❣ ♥❣÷í✐ ✤➛✉ t✐➯♥ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ sè t❤❡♦ q✉❛♥ ✤✐➸♠ ❤✐➺♥ ✤↕✐ ♥❤÷ ♥❣➔② ♥❛②✳ ◗✉❛♥ ✤✐➸♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ♥➔② ❝❤➾ ①➨t sü ❤ỉ✐ tư ❝õ❛ ♠ët ❝❤✉é✐ sè t❤ỉ♥❣ q✉❛ sü ❤ë✐ tư ❝õ❛ ♠ët ❞➣② tê♥❣ r✐➯♥❣ (sn)✳ ❚rð ❧↕✐ ✈➜♥ ✤➲ ♥➔② t❛ ①➨t ❝❤✉é✐ +∞ (−1)n = − + − + , n=0 ❝â ❞➣② tê♥❣ r✐➯♥❣ (sn ) = 1, 0, 1, 0, 1, ổ tỗ t ợ õ ởt ỵ tữ ữủ tr số ❤å❝ sn = s0 + s1 + + sn n+1 ❇ð✐ ✈➻ sn = 12 [1 + (−1)n]✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ (n + 1) + 12 [1 + (−1)n ] 1 + (−1)n sn = = + 2(n + 1) 4(n + 1) ❚❤❡♦ ♥❣❤➽❛ ♥➔② ❞➣② sn ❤ë✐ tư tỵ✐ ❣✐→ trà ữỡ tr ỵ r ♥❤÷ t❤➳ ❞➝♥ ✤➳♥ 1 − + − + = ❚ø ✤â✱ ♥❣÷í✐ t❛ ✤÷❛ r❛∞❦❤→✐ ♥✐➺♠ ❝â t❤û ♥❣❤✐➺♠ ♥❤÷ s❛✉✿ ❉➣② tê♥❣ r✐➯♥❣ (sn) ❤♦➦❝ ❝❤✉é✐ an ✤÷đ❝ ❣å✐ ❧➔ ✧❤ë✐ tư✧ tỵ✐ ✧❣✐ỵ✐ ❤↕♥✧✱ ❤♦➦❝ n=0 ❝â tê♥❣ ❧➔ s ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞➣② sn = s0 + s1 + + sn n+1 ❤ë✐ tö+∞ ✤➳♥ s✳ ❚➼♥❤ t❤➼❝❤ ❤ñ♣ ❝õ❛ ❦❤→✐ ♥✐➺♠ ♥➔② ♠✐♥❤ ❝❤ù♥❣ r➡♥❣ ❝❤✉é✐ (−1)n trð t❤➔♥❤ ♠ët ❝❤✉é✐ ❤ë✐ tö ✧t❤❡♦ ♥❣❤➽❛ ♠ỵ✐ ✧ ✈ỵ✐ n=0 tê♥❣ ❜➡♥❣ 12 ✳ ❈â ữ ỵ ữ r ợ ữ s (i) (sn ) tử t tổ tữớ tợ s ụ ữ tử ✤➳♥ s t❤❡♦ ♥❣❤➽❛ ♠ỵ✐✳ +∞ +∞ (ii) ◆➳✉ ❤❛✐ ❝❤✉é✐ ❤ë✐ tư t❤❡♦ ♥❣❤➽❛ ❝ơ an = A ✈➔ bn = B t❤➻ n=0 +∞ n=0 +∞ ❝❤✉é✐ t➼❝❤ cn ≡ (a0bn + a1bn−1 + + anb0) ❦❤æ♥❣ ♥❤➜t t❤✐➳t n=0 n=0 ❤ë✐ tö t❤❡♦ ♥❣❤➽❛ ♥➔②✳ ❚❤➳ ♥❤÷♥❣ t❛ t❤➜② r➡♥❣ C0 + C1 + + Cn → AB; n+1 n +∞ ✈ỵ✐ Cn = ck ✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❝❤✉é✐ t➼❝❤ Cn ❧↕✐ ❤ë✐ tư n=0 k=0 t❤❡♦ ♥❣❤➽❛ ♠ỵ✐✳ ◆❣♦➔✐ ❞↕♥❣ tr✉♥❣ ❜➻♥❤ số tr ụ ủ ỵ ự ✤➳♥ ♠ët q✉→ tr➻♥❤ ❦❤→❝ ❝â t❤➸ ❞ò♥❣ t❤❛② t❤➳ ♠ët ❞↕♥❣ ❦❤→✐ ♥✐➺♠ ❤ë✐ tö✳ ❱✐➺❝ →♣ ❞ö♥❣ C1−q✉→ tr➻♥❤ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ❝❤✉é✐ ❋♦✉r✐❡r ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❋❡❥❡✬r✱ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤✐➲✉ ✹ ❦✐➺♥ ✤➸ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) ❤ë✐ tö t↕✐ ✤✐➸♠ x0✳ ợ ố t ỵ tt ộ ý ộ rr t ỵ t❤✉②➳t ❝❤é✐ ♣❤➙♥ ❦ý ✈➔ ù♥❣ ❞ö♥❣✑✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ỵ tt ộ ý C1q tr ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤✉é✐ ❋♦✉r✐❡r ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ự ự ỵ tt ộ ý ỵ tt ộ rr Pữỡ ❝ù✉ ❚r❛ ♠↕♥❣ t➻♠ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤ ✈➔ tê♥❣ ủ tự ỵ ữợ ữớ ữợ ỹ õ õ t r ởt õ tố ỵ tt ❝❤✉é✐ ♣❤➙♥ ❦ý✱ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❝→❝ C1−q✉→ tr➻♥❤ ✈➔ ù♥❣ C1q tr tr ự ỵ tt ❝❤✉é✐ ❋♦✉r✐❡r✳ ✺ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ k ≥ t❛ ❧✉æ♥ ❝â n+k n+k sυ (bυ − bυ+1 ) − sn bn+1 + sn+k bn+k+1 aυ b υ = υ=n+1 ❈❤ù♥❣ ♠✐♥❤✳ υ=n+1 ❚ø ❝→❝❤ ❦➼ ❤✐➺✉ tr➯♥ t❛ ❝â aυ = sυ − sυ−1 ; ✈ỵ✐ ♠å✐ υ = 0, 1, 2, ❑❤✐ υ = 0✱ t q ữợ s1 = õ n+k a bυ = an+1 bn+1 + an+2 bn+2 + + an+k bn+k υ=n+1 = (sn+1 − sn )bn+1 + (sn+2 − sn+1 )bn+2 + + (sn+k − sn+k−1 )bn+k = (sn+1 bn+1 − sn+1 bn+2 ) + (sn+2 bn+2 − sn+2 bn+3 ) + +(sn+k bn+k − sn+k bn+k+1 ) − sn bn+1 + sn+k bn+k n+k sυ (bυ − bυ+1 ) − sn bn+1 + sn+k bn+k+1 = =n+1 ỵ ữủ ự ỵ ộ + an ợ tờ sn ❧➔ C1 −❦❤↔ tê♥❣ t❤➻ n=1 an = o(n) ✈➔ sn = o(n) ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ ❝❤✉é✐ ✤➣ ❝❤♦ ❧➔ C1−❦❤↔ tê♥❣ ♥➯♥ ❞➣② s❛✉ ✤➙② ❤ë✐ tö sn n+k k ❇ð✐ ✈➻ n ∼ = n+1 = s0 + s1 + + sn n+k k ♥➯♥ ❞➣② s0 + s1 + + sn n+k k ❝ơ♥❣ ❤ë✐ tư ✤➳♥ ❝ò♥❣ ♠ët ❣✐ỵ✐ ❤↕♥✳ ❉♦ ✤â✱ ♣❤➛♥ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❤❛✐ sn ❧➔ ♠ët ❞➣② ❤ë✐ tö ✈➲ 0✳ ❱➻ n1 ∼ n ❦➨♦ t❤❡♦ sn = o(n)✳ t❤÷ì♥❣ n+1 ✸✼ ❚ø ✤â t❛ ❝â s n = o(n), sn = o(n), an = o(n) ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✤à♥❤ ỵ ỵ ộ + an ợ tờ r✐➯♥❣ (sn ) ❧➔ C1 −❦❤↔ tê♥❣ ✈ỵ✐ n=1 tê♥❣ s✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❧➔ ❝❤✉é✐ +∞ ν=0 ✭✷✳✶✮ aν ν+1 ❤ë✐ tư ✈➔ ♣❤➛♥ ❞÷ ❝õ❛ ❝❤✉é✐ Rn = an+2 an+1 + + (n = 1, 2, ) n+2 n+3 t❤ä❛ ♠➣♥ sn + (n + 1)Rn → s ◆➳✉ σn ❧➔ tê♥❣ r✐➯♥❣ ❝õ❛ ❝❤✉é✐ ✭✷✳✷✮ ✭✷✳✶✮ ✈➔ σ ❧➔ tê♥❣ ❝õ❛ ♥â t❤➻ tø ✭✷✳✷✮ t❛ ♥❤➟♥ ✤÷đ❝ s − sn − (n + 1)(σ − σn ) → ✭✷✳✸✮ +∞ ●✐↔ sû ❝❤✉é✐ an ❧➔ C1−❦❤↔ tê♥❣✳ ❚ø aυ = sυ − s1 n=1 ỵ t õ ự ♠✐♥❤✳ n+p n+p sn sυ sn+p aυ =− + + , υ + n + (υ + 1)(υ + 2) n + p + υ=n+1 υ=n+1 ✈➔ tø sυ = s υ − s υ−1✱ t✐➳♣ tö❝ ỵ ởt ỳ t tờ tr➯♥ trð t❤➔♥❤ n+p n+p aυ sn sn sυ =− − +2 υ+1 n + (n + 2)(n + 3) (υ + 1)(υ + 2)(υ + 3) υ=n+1 υ=n+1 ✸✽ + sn+p s n+p + n + p + (n + p + 2)(n + p + 3) ❑❤✐ n → +∞✱ t➜t ❝↔ ♥➠♠ sè ❤↕♥❣ tr♦♥❣ ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ✤➲✉ ❞➛♥ ✤➳♥ 0✱ ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ p✱ tø ❣✐↔ t❤✐➳t ộ C1 tờ t ỵ 2.3 s r❛ sn = 0(n) ✈➔ sn = 0(n)✳ ❚÷ì♥❣ tü ❦❤✐ ❝❤♦ n ✈➔ p → +∞✱ t❛ ❝ô♥❣ ❝â +∞ sυ sn + 2(n + 2) sn + (n + 2)Rn = − n+3 (υ + 1)(υ + 2)(υ + 3) υ=n+1 ✤➲✉ ❞➛♥ ✤➳♥ s✱ t❤❡♦ ❬9✱ 221❪ ✈➻ ns+υ → s✳ ❉♦ ✤â✱ t❛ ❝â ✭✷✳✷✮ ❦❤✐ Rn → ❇➙② ❣✐í t❛ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐✱ ❝→❝ ❜✐➸✉ t❤ù❝ ✭✷✳✶✮ ✈➔ ✭✷✳✷✮ ✤÷đ❝ t❤ä❛ ♠➣♥✱ +∞ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤✉é✐ an ❧➔ C1−❦❤↔ tê♥❣✳ ◆➳✉ t❛ ❦➼ ❤✐➺✉ τn ❧➔ ✈➳ n=1 tr→✐ ❝õ❛ ❜✐➸✉ t❤ù❝ ✭✷✳✷✮ t❤➻ t❛ ❝â τn+1 − τn = sn+1 + (n + 2)Rn+1 − sn − (n + 1)Rn = an+1 + (n + 2)Rn+1 − (n − 1)Rn = Rn + an+1 + (n + 2)(Rn+1 − Rn ) = Rn +an+1 +(n+2) ✈➔ an+2 an+3 an+1 an+2 + + − − − n+3 n+4 n+2 n+3 = Rn , τn = sn + (n + 1)(τn+1 − τn ) ❚ø ✤â s✉② r❛ sn = 2τn − [(n + 1)τn+1 − nτn ] ❉♦ ✤â s0 + s1 + + sn τ0 + τ1 + + τn =2 − τn+1 n+1 n+1 ◆❤÷♥❣ τn → s ✈➻ ✈➟② ❞➣② (sn) ❧➔ C1−❣✐ỵ✐ ❤↕♥ ✈ỵ✐ ❣✐→ trà s✳ ❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✸✾ ✷✳✹✳ ⑩♣ ❞ö♥❣ ❝õ❛ C1−tê♥❣ ố ợ ộ rr ộ ữủ ✷✳✷✳ ❈❤✉é✐ ❤➔♠ ❧÷đ♥❣ ❣✐→❝ ❧➔ ❝❤✉é✐ ❤➔♠ ❝â ❞↕♥❣ +∞ a0 + (an cos nx + bn sin nx) n=1 ✭✷✳✹✮ tr♦♥❣ ✤â a0, an, bn, n = 1, 2, số tỹ rữợ t t ú ỵ r ộ tử õ tê♥❣ ❧➔ f (x) t❤➻ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ 2π ✳ ❱➻ t❤➳ s❛✉ ✤➙② t❛ ❝❤➾ ❝➛♥ ①➨t ❤➔♠ sè tr➯♥ ♠ët ✤♦↕♥ ❝â ✤ë ❞➔✐ ❜➡♥❣ 2π✳ ❈❤➥♥❣ ❤↕♥ ❧➔ [−π, +π]✳ ●✐↔ sû ❝❤✉é✐ ❤➔♠ ❧÷đ♥❣ ❣✐→❝ ✭✷✳✹✮ ❤ë✐ tư ✤➲✉ tr➯♥ ✤♦↕♥ [−π, +π] ✈➔ +∞ a0 f (x) = + (an cos nx + bn sin nx); x ∈ [✲π ✱π]✳ n=1 ✭✷✳✺✮ ❚ø ✭✷✳✺✮ t❛ ♥❤➙♥ ❝↔ ❤❛✐ ✈➳ ✈ỵ✐ cos kx s❛✉ ✤â ❧➜② t➼❝❤ ♣❤➙♥ tø♥❣ số tr [, +] ợ ữ ỵ sỷ ❞ö♥❣ ❝→❝ ✤➥♥❣ t❤ù❝ s❛✉ π cos kx cos nxdx = 0 π −π π sin kx sin nxdx = 0 π −π ♥➳✉ k = n ♥➳✉ k = n ♥➳✉ k = n ♥➳✉ k = n π sin kx cos nxdx = −π ✹✵ ✈ỵ✐ ♠å✐ k, n ❚❛ ❝â π a0 = π f (x)dx −π π ak = π f (x) cos kxdx; ✈ỵ✐ k = 1, 2, f (x) sin kxdx; ✈ỵ✐ k = 1, 2, −π ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â π bk = π −π ✷✳✹✳✷✳ ❈❤✉é✐ ❋♦✉r✐❡r ●✐↔ sû f (x) ❧➔ ♠ët ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [−π, +π] ✈➔ t✉➔♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦ý 2π✳ ❑❤✐ ✤â✱ ❝→❝ ❤➺ sè an, bn ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ π ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ an = π f (x) cos nxdx; n = 1, 2, −π π bn = π f (x) sin nxdx; n = 1, 2, −π ✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x)✱ ❝á♥ ❝❤✉é✐ ❧÷đ♥❣ ❣✐→❝ +∞ a0 + (an cos nx + bn sin nx) n=1 ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x)✳ ✷✳✹✳✸✳ ⑩♣ ❞ư♥❣ ❝õ❛ C1 −tê♥❣ ✤è✐ ✈ỵ✐ ❝❤✉é✐ ❋♦✉r✐❡r ❱✐➺❝ →♣ ❞ö♥❣ C1−q✉→ tr➻♥❤ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤✉é✐ ❋♦✉r✐❡r ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❋❡❥❡✬r✱ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤✐➲✉ ❦✐➺♥ ✤➸ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) ❤ë✐ tö t x0 ỵ r ữủ t ữ s ỵ ỵ r ♠ët ❤➔♠ f (x) ❧➔ ❦❤↔ t➼❝❤ tr♦♥❣ ❦❤♦↔♥❣ ≤ x ≤ 2π ✱ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ 2π ✈➔ ❣✐ỵ✐ ❤↕♥ lim [f (x0 + 2t) + f ✭x0 − 2t✮] = s(x0 ) t→0 tỗ t t ộ rr ổ C1 −❦❤↔ tê♥❣ t↕✐ ✤✐➸♠ x0 ✈ỵ✐ tê♥❣ s(x0 )✳ ●å✐ ❈❤ù♥❣ ♠✐♥❤✳ +∞ a0 + (an cos nx0 + bn sin nx0 ) n=1 ❧➔ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) t↕✐ ✤✐➸♠ x0✳ ❚❛ ❝â a0 sn (x0 ) = + ❚r♦♥❣ ✤â n (aυ cos υx0 + bυ sin υx0 ) υ=1 2π a0 = 2π f (t)dt; aυ cos υx0 +bυ sin υx0 = 2π 2π f (t) cos υtdt cos υx0 + π π = π ❉♦ ✤â 2π sn (x0 ) = 2π = 2π f (t) sin υtdt sin υx0 2π f (t) cos υ(t − x0 )dt 2π f (t)dt+ π 2π f (t) cos(t − x0 )dt + + π f (t) cos n(t − x0 )dt 2π + cos(t − x0 ) + cos 2(t − x0 ) + + cos n(t − x0 ) dt ❚❤❡♦ ❝→❝❤ ❜✐➳♥ ✤ê✐ ❝õ❛ ✈➼ ❞ö 2✱ ♠ö❝ 2.2 ð tr➯♥ t❤➻ t − x0 sin(2n + 1) +cos(t−x0 )+cos 2(t−x0 )+ +cos n(t−x0 ) = ✭✷✳✻✮ t − x0 2 sin f (t) ✹✷ ❉♦ ❤➔♠ f (x) t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ 2π ❞♦ ❞â t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ t➼❝❤ ♣❤➙♥✱ ✈ỵ✐ c ❧➔ ♠ët sè ❜➜t ❦➻ 2π c+2π f (t)dt = f (t)dt = c β 2π f (c + t)dt ❚ø ✭✷✳✻✮✱ t❛ ❝â s(x0 ) = 2π sin(2n + 1) f (x0 + t) sin = = = ❱➟② t f (t)dt α+2π t dt t t sin(2n + 1) 2π dt + f (x + t) dt f (x0 + t) t t 2π 2π π sin sin 2 t t sin(2n + 1) sin(2n + 1) π −2π dt − dt f (x0 + t) f (x0 − t) t − x0 t 2π 2π −π sin sin 2 π π 1 sin(2n + 1)t sin(2n + 1)t dt + dt f (x0 + 2t) f (x0 − 2t) 2π sin t 2π sin t π 21 sin(2n + 1)t [f (x0 + 2t) + f (x0 − 2t)] dt; π0 sin t sin(2n + 1) π = f (t)dt = α 2π ✈➔ β+2π sn −sn (x0 ) = π π sin(2n + 1)t [f (x0 + 2t) + f ✭x0 − 2t✮] dt; sin t ✈ỵ✐ n = 0, 1, 2, ❉♦ ✤â ✈ỵ✐ n = 0, 1, 2, t❛ ❝â s0 + s1 + sn−1 = ❑❤✐ ✤â sin t + sin 2t + + sin(2n − 1)t [f (x0 + 2t) + f ✭x0 − 2t✮] dt π sin t ✹✸ ợ ộ t = k t ữủ sin t + sin 2t + + sin(2n − 1)t 2sin2 t + sin t sin 2t + + sin t sin(2n − 1)t = sin t = [(1 − cos 2t) + (cos 2t − cos 4t) + + (cos(2n − 2)t − cos 2nt)] sin t sin2 nt = (1 − cos 2nt) = sin t sin t nt ✰ ✈ỵ✐ t = kπ t❛ t❤➜② sin → ❦❤✐ t → kπ ✳ ❉♦ ✤â sin t π 2 s0 + s1 + + sn−1 sin nt σn+1 = = [f (x0 +2t)+f ✭x0 −2t✮] dt n nπ sin t ❉♦ õ tỗ t ợ lim [f (x0 + 2t) + f ✭x0 − 2t✮] = s(x0 ) = s t→0 t❤➻ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤à♥❤ ỵ r n s t t t sin nt sin t dt = nπ n 1)t ữợ t sin(2ν ✈ỵ✐ ♠é✐ sè ❤↕♥❣ ❝õ❛ ♥â t❤➻ sin t ν=1 t➼❝❤ ♣❤➙♥ tø ✤➳♥ π2 ✤➲✉ ♥❤➟♥ ❣✐→ trà π2 ✳ ❇ð✐ ✈➻ sin(2ν − 1)t = + cos 2t + cos 4t + + cos 2(ν − 1)t sin t ♥➯♥ t❛ ❝â t❤➸ ✈✐➳t s= nπ π sin nt s sin t ✹✹ dt ❉♦ ✤â π 2 σn−1 − s = nπ sin nt [f (x0 + 2t) + f (x0 − 2t) − s] sin t dt ❚ø ❣✐↔ t❤✐➳t✱ ❜✐➸✉ t❤ù❝ tr♦♥❣ ❞➜✉ ♥❣♦➦❝ ✈✉ỉ♥❣ ❧➔ ❞➛♥ tỵ✐ ❦❤✐ t → +0✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ σn−1 ❤❛② σn → s✱ t❛ s➩ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ s❛✉ (t) ữợ t tứ ✤➳♥ π 2 nπ π ✈➔ lim ϕ(t) = t❤➻ t→+0 2 sin nt ϕ(t) sin t dt → 0, ✈ỵ✐ n → ∞ ❇ð✐ ✈➻ ϕ(t) → 0, t❛ ❝â t❤➸ ❝❤♦ δ < ợ > trữợ s ε |ϕ(t)| < ; ✈ỵ✐ ♠å✐ t ∈ (0, δ] ❦❤✐ ✤â nπ π sin nt ϕ(t) sin t ε dt ≤ nπ π sin nt sin t dt < ε tø t➼❝❤ ♣❤➙♥ ❝✉è✐ ❝ò♥❣ ❤➔♠ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ❤➔♠ ❞÷ì♥❣ ✈➔ ❞♦ ✤â t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ❜➨ ❤ì♥ t➼❝❤ ♣❤➙♥ ❝õ❛ ♠ët ✈➔✐ ❤➔♠ ❦❤✐ t❛ ❧➜② t➼❝❤ ♣❤➙♥ tø ✤➳♥ π2 ✳ sỷ tỗ t số M s (t) < M ✈ỵ✐ < t < π2 ✳ ❑❤✐ ✤â π 2 nπ sin nt ϕ(t) sin t dt ≤ ✹✺ 2M π nπ sin2 δ ❉♦ ✤â ❝â t❤➸ ❝❤å♥ n0 ✤õ ❧ỵ♥ s❛♦ ❝❤♦ ❜✐➸✉ t❤ù❝ ♥➔② ♥❤ä ❤ì♥ 2ε ✈ỵ✐ ♠å✐ n > n0 ❑❤✐ ✤â✱ t❛ ❝â |σn−1 − s(x0 )| < ε ❞♦ ✤â σn−1 → s✳ ❚❤❡♦ t❤❡♦ ❬ 9✱ 267❪ t❤➻ t❛ ❝ô♥❣ ❝â σn → s✳ ❍❛② ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ❧➔ C1−❦❤↔ tờ ợ tờ s(x0) ỵ r ữủ ự ①♦♥❣✳ ❱➼ ❞ö ✷✳✹✳ a) ❳➨t ❤➔♠ f (x) = 0 ♥➳✉ −π ≤ x ≤ ♥➳✉ ≤ x ≤ π x ❉➵ t❤➜② f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ 2π✱ ✤ì♥ ✤✐➺✉ tø♥❣ ❦❤ó❝ ✈➔ ❜à ❝❤➦♥ tr➯♥ ✤♦↕♥ [ − π, π]✳ ❚❛ ❝â π a0 = π f (x)dx = π ; −π an = π π f (x) cos nxdx = x cos kxdx π −π π −π π x sin kx cos kx π = sin kxdx = + π k k kπ k 0 0 ♥➳✉ k ❝❤➤♥ = −2 ♥➳✉ k ❧➫ π.k ✈➔ π bn = π π f (x) sin kxdx = π −π x cos kx = π k x sin kxdx −π π + k ✹✻ π cos kxdx = − π cos kx|π0 kπ π −1 = 1k k ♥➳✉ k ❝❤➤♥ ♥➳✉ k ❧➫ ❉♦ ✤â ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) ❧➔ f (x) = π − π cos x cos 3x cos 5x sin x sin 2x sin 3x + + + + + + + 2 ❝❤✉é✐ ♥➔② ❤ë✐ tö t↕✐ ♠å✐ ✤✐➸♠ tr➯♥ ✤♦↕♥ [ − π, π]✱ trø ❝→❝ ✤✐➸♠ x = ±π✳ ❚↕✐ ❝→❝ ✤✐➸♠ ♥➔② tê♥❣ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) ❜➡♥❣ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❣✐ỵ✐ ❤↕♥ ♣❤↔✐ ✈➔ ❣✐ỵ✐ ❤↕♥ tr→✐ ❝õ❛ ❤➔♠ t↕✐ ❝→❝ ✤✐➸♠ ✤â✱ ♥❣❤➽❛ ❧➔ ❜➡♥❣ π +2 = π2 b) ❍➔♠ f (x) t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ 2π ✱ ❜✐➳t f (x) = x ✈ỵ✐ ♠å✐ x ∈ [✵❀✷π]✳ ❚❛ ❝â 2π a0 = π f (x)dx = 2π; an = π ❉♦ ✤â ❝❤✉é✐ ❋♦✉r✐❡r 2π f (x) cos nxdx = 0; 2π bn = f (x) sin nxdx = − π0 n ❝õ❛ ❤➔♠ f (x) ❧➔ s(x) = π − 2(sin x + sin 2x + ) ỵ r ❝❤✉é✐ s(x) ❤ë✐ tư ✤➳♥ f (x) ✈ỵ✐ ♠å✐ x = 2kπ, k ∈ Z ❚↕✐ ❝→❝ ✤✐➸♠ x = 2kπ, s(x) ❤ë✐ tö ✤➳♥ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❣✐ỵ✐ ❤↕♥ tr→✐ ✈➔ ❣✐ỵ✐ ❤↕♥ ♣❤↔✐ t↕✐ ❝→❝ ✤✐➸♠ ✤â✱ ♥❣❤➽❛ ❧➔ ❜➡♥❣ 2π 2+ = π ❍➺ q✉↔ ✷✳✶✳ ◆➳✉ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr♦♥❣ ❦❤♦↔♥❣ ≤ x ≤ 2π ✈➔ ❝â f (0) = f (2π)✱ t❤➻ ❝❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (x) ❧➔ C1 −❦❤↔ tê♥❣ ✈ỵ✐ tê♥❣ f (x)✱ ✈ỵ✐ ♠å✐ x✳ ✹✼ ❍➺ q✉↔ ✷✳✷✳ ◆➳✉ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr♦♥❣ ❦❤♦↔♥❣ ≤ x ≤ 2π ❀ f (0) = f (2π) ✈➔ ❧➔ C1 −❦❤↔ tê♥❣ t❤➻ t❛ ❝â t❤➸ t❤✐➳t ❧➟♣ ♠ët ❞➣② ❝→❝ ❤➔♠ σn (x) ✤➲✉ t✐➳♥ ❞➛♥ ✤➳♥ ❤➔♠ f (x) ✈ỵ✐ ♠å✐ x r ợ > trữợ t ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷đ❝ sè tü ♥❤✐➯♥ N s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n > N t❛ ✤➲✉ ❝â |σn (x) − f (x)| < ε ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ |σn (x) − f (x)| < ε ✣➦t ϕ(t) = ϕ(t, x) = [f (x + 2t) − f (t)] + [f (x − 2t) − f (x)]; tø ❣✐↔ t❤✐➳t ❤➔♠ f (x) ❧➔ t✉➛♥ ❤♦➔♥ ✈➔ ❧✐➯♥ tö❝ ✤➲✉ t↕✐ x ợ >0 trữợ t õ t❤➸ ❝❤å♥ δ > s❛♦ ❝❤♦ |f (x ± 2t) f (x)| < ợ x ữ ❜✐➳t δ sin nt ϕ(t) sin t nπ ε ε dt < ❍ì♥ ♥ú❛✱ ❤➔♠ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥✱ ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ ✈➔ ❜à ❝❤➦♥ ❤❛② |f (x)| < K ợ x tự t õ ữợ ✤➙② ✈ỵ✐ ♠å✐ t ✈➔ x |ϕ(t)| = |ϕ(t, x)| < 2K ✈➔ ❞♦ ✤â δ sin nt ϕ(t) sin t nπ dt < 2K n sin2 δ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ r r tỗ t số ữỡ N s ❝❤♦ ❜✐➸✉ t❤ù❝ tr➯♥ < 2ε ✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ n t❛ ❝â |σn(x) − s| < ε✳ ✹✽ ❑➌❚ ▲❯❾◆ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❦❤â❛ ỵ tt ộ ý ự õ ❧✉➟♥ ✤➣ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ♥❤÷ s❛✉ ❈❤÷ì♥❣ 1✳ ❚r➻♥❤ ❜➔② ❤➺ t❤è♥❣ ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ ❦❤→✐ ♥✐➺♠ ❝ơ♥❣ ♥❤÷ ❝→❝ ✈➼ ❞ư ✈➲ ❝❤✉é✐❀ ❝→❝ ỵ sỹ tử ộ số ộ ộ ụ tứ ữỡ ữ r ỵ ✈➲ sü ♣❤➙♥ ❦ý ❝õ❛ ❝❤✉é✐❀ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ✈➲ C1−❦❤↔ tê♥❣ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ C1−❦❤↔ tê♥❣ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤✉é✐ ❋♦✉r✐❡r✳ ❉♦ ♥➠♥❣ ❧ü❝ ✈➔ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟②✱ t→❝ ❣✐↔ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ỡ ỳ ỵ õ õ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ✣ù❝ ▲♦♥❣ ✲ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ✲ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥✱●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤✱ t➟♣ ✷✱ ◆❳❇ ✣↕✐ ❍å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✱ ✭✷✵✵✷✮✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✲ ✣➟✉ ❚❤➳ ❈➜♣ ✲ ❇ò✐ ✣➢❝ ❚➢❝✱ ❚♦→♥ ❝❛♦ ❝➜♣✱ ◆❳❇ ❑❤♦❛ ❍å❝ ✈➔ ❑➽ ❚❤✉➟t✱ ✭✶✾✾✽✮✳ ❬❇❪ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ữợ trs r 1+ m(m − 1) m x+ x + 1.2 ❏♦✉r♥❛❧ ❢☎ ✉r ❞✐❡ r❡✐♥❡ ✉♥❞ ❛♥❣❡✇❛♥❞t❡ ▼❛t❤❡♠❛t✐❦✱ ❱♦❧✳ ✶✱ ♣♣✳ 311 − 339.(1826) ❬✹❪ ❆✳ ▲✳ ❈❛✉❝❤②✱ ❈♦✉rs ❞✬❛♥❛♥❧②s❡ ❞❡ ❆♥❛♥❧②s❡ ❛✐❣➨❜r✐q✉❡✳ P❛r✐s ✭✶✽✷✶✮✳ ❧✬£❝♦❧❡ ♣♦❧②t❡❝❤♥✐q✉❡✱ P❛rt ■✳ ❬✺❪ ▲✳❊✉❧❡r✱ ■♥tr♦❞✉❝t✐♦ ✐♥ ❛♥❛♥❧②s✐♥ ✐♥❢✐♥✐t♦r✉♠✱ ▲❛✉s❛♥♥❡✭✶✼✹✽✮✳ ❬✻❪ ▲✳ ❊✉❧❡r✱ ■♥st✐t✉t✐♦♥❡s ❝❛❧❝✉❧✐ ❞✐❢❢❡r❡♥t✐❛❧✐s ❝✉♠ ❡❥✉s ✉s✉ ✐♥ ❛♥❛❧②s✐ ✐♥❢✐♥✐t♦r✉♠ ❛❝ ❞♦❝tr✐♥❛ s❡r✐❡r✉♠✱ ❇❡r❧✐♥✭✶✼✺✺✮✳ ❬✼❪ ▲✳ ❊✉❧❡r✱ ■♥st✐t✉t✐♦♥❡s ❝❛❧❝✉❧✐ ✐♥t❡❣r❛❧✐s✱ ❙t✳ P❡t❡rs❜✉r❣✭✶✼✻✽ ✲ ✶✼✻✾✮✳ ❬✽❪ ❑✳ ❋✳ ●❛✉ss✱ ❉✐sq✉✐s✐t✐♦♥❡s ❣❡♥❡r❛❧❡s ❝✐r❝❛ s❡r✐❡♠ ✐♥❢✐♥✐t❛♠ 1+ α.β α(α + 1).β(β + 1) x+ x + etc 1.γ 1.2.γ(γ + 1) ●❙tt✐♥❣❡♥✭✶✽✶✷✮✳ ❬✾❪ ❉❘✳ ❑♦♥r❛❞ ❑♥♦♣♣✱ ❘❡♣r✐♥t❡❞✭✶✾✺✹✮✳ ❚❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ✐♥❢✐♥✐t❡ s❡r✐❡s✱ ✺✶