Bài tập Toán A2 TNKT tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩnh vực kinh tế...
❇⑨■ ❚❾P ❱⑨ ❈⑩❈ ❱❻◆ ✣➋ ❈❺◆ ❚❍❷❖ ▲❯❾◆ ❈❤÷ì♥❣ ✶✳ ❙è t❤ü❝ ✈➔ ❣✐ỵ✐ ❤↕♥ ❞➣② sè ♣ ❈➙✉ ❤ä✐ t❤↔♦ ❧✉➟♥ ✶✳ ◆➯✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❞➣② sè ❤ë✐ tö✱ ❞➣② sè ♣❤➙♥ ❦ý ✷✳ ◆➯✉ ữỡ t ợ số ❚➻♠ ♠ët sè ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ❞➣② ❤ë✐ tö✱ ♣❤➙♥ ❦ý✳ ♣ ❇➔✐ t➟♣ ❇➔✐ ✶✳ ❚➻♠ ✐♥❢✱ s✉♣ ❝õ❛ ❝→❝ t➟♣ s❛✉ 1.1.A = { : n = 1, 2, } 2n n : n = 1, 2, } n+2 −π π , ]} 1.4.D = {cos xex : x ∈ [ 2 1.2.B = { 1.3.C = {x sin x : x ∈ (0, π]} ❇➔✐ ✷✳ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ✈➲ ❝→❝ ❣✐ỵ✐ ❤↕♥ s❛✉ ✷✳✶✳ α n n→+∞ =0 ✷✳✷✳ nk n n→+∞ a = 0, (a > 1; k > 0) ✷✳✸✳ lim lim lim √ n ✈ỵ✐ α > 0✳ a = 1, (a > 0) n→+∞ ✷✳✹✳ ✷✳✺✳ an n→+∞ n! lim lim √ n = 0, (a > 0) np = 1, ✈ỵ✐ ♠å✐ p n→+∞ ✷✳✻✳ lim q n = 0, n→+∞ ♥➳✉ |q| < 1; limn+ q n = 1, ổ tỗ t tr tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐✳ ✼✮✷✳✼✳ lim (1 + n1 )n = e, n→±∞ ✷✳✽✳ lim n→+∞ ✷✳✾✳ n √ n n! np n (n+a) n→∞ lim = e, = (∀p > 0, a > 0) ✶ ♥➳✉ q = ú ỵ n n n (1+p) lim = 0, p > 0, α ❜➜t ❦ý ❚❤ỉ♥❣ t❤÷í♥❣✱ ✤➸ t➻♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② sè t❛ ❞ü❛ ✈➔♦ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❝→❝ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ❜➔✐ t➟♣ ✷✮✳ ❇➔✐ ✸✳ ❚➻♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ ❞➣② s❛✉ ✸✳✶✳ lim ( 2.1 + 2.3 lim ( 12 + 22 n→∞ ✸✳✷✳ n→+∞ + + + 23 n n(n+1) ) + + 2n−1 2n ) √ √ √ √ n lim ( 2) ✸✳✸✳ n→∞ lim [(1 − ✸✳✹✳ n→∞ 22 )(1 32 ) (1 − − lim [(1 − 31 )(1 − 16 ) (1 − ✸✳✺✳ n→∞ 3 n2 )] n(n+1) )] 3 −1 n −1 lim ( 223 −1 +1 33 +1 n3 +1 ) ✸✳✻✳ n→∞ lim ( 1+1 + ✸✳✼✳ 1+23 + + 1+n3 ) lim [(1 + 12 )(1 + 22 ) (1 + 2x lim [ cos x−cos + cos 2x−cos 3x n→∞ ✸✳✽✳ n→∞ ✸✳✾✳ n→∞ ✸✳✶✵✳ lim ( √11.2 + n→∞ ✸✳✶✶✳ √1 2.3 + + √ 2n )] + + ) n(n+1) √ √ lim ( n + − n) n→∞ ✸✳✶✷✳ lim √ n n2 2n + n n→∞ ✸✳✶✸✳ lim n→∞ ✸✳✶✹✳ n sin n n2 +1 lim nq n (|q| < 1) n→∞ ✸✳✶✺✳ lim ( sin2 + n→∞ ✸✳✶✻✳ 1! lim ( cos 1.2 + n→∞ sin 22 + + cos 2! 2.3 sin n 2n ) + + cos n! n(n+1) ) ✷ cos nx−cos(n+1)x ] n ✸✳✶✼✳ 22 lim (1 + n→∞ + 32 + + n2 ) ❇➔✐ ✹✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❝→❝ ❞➣② sè s❛✉ ♣❤➙♥ ❦ý ✭an ❧➔ ♣❤➛♥ tû tê♥❣ q✉→t ❝õ❛ ❞➣② sè ✭an ✮✮ ✹✳✶✳ an = (−1)n ✹✳✷✳ an = + 12 + 13 + + ✹✳✸✳ an = ln ✹✳✹✳ an = √1 ln ✹✳✺✳ an = n ✹✳✻✳ an = ✹✳✼✳ an = (−1)n + ln + + + + √1 ln n ln n √1 n ln n + + n+(−1)n n n+2 n ❇➔✐ ✺✳ ❈❤ù♥❣ ♠✐♥❤ ❞➣② s❛✉ ❤ë✐ tö ✺✳✶✳ xn = + 12 + 14 + + ✺✳✷✳ xn = ✺✳✸✳ xn = sin + sin 22 ✺✳✹✳ xn = + 22 + ✺✳✺✳ xn = + √1 2.3 ✺✳✻✳ < xn < 2, xn+1 = + xn ✺✳✼✳ xn = 2.5 ✺✳✽✳ xn = sin 1+1 ✺✳✾ xn = ✺✳✶✵✳ 2+ √1 1.2 2+3 xn = + 2+ 3.6 + + + + 32 + + n2 + + √ + + n(n+1) (n+1)(n+4) + + sin n 1+n3 22 +32 + + 2n +3n √ n2 +1 + + √ sin n 2n sin 1+23 + + 2n √ n2 +n ✸ ✭♥ ❞➜✉ ❝➠♥✮ ❇⑨■ ❚❾P ❱⑨ ❈⑩❈ ❱❻◆ ✣➋ ❈❺◆ ❚❍❷❖ ▲❯❾◆ ❈❤÷ì♥❣ ✷✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ sè ✈➔ ❤➔♠ sè ❧✐➯♥ tư❝ ♣ ❈➙✉ ❤ä✐ t❤↔♦ ❧✉➟♥ ✶✳ ◆➯✉ ❝→❝ ♣❤÷ì♥❣ t ợ ởt số ú ỵ trữớ ủ ợ t ổ ỹ ợ ổ ỹ ỵ trữ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧✐➯♥ tư❝✱ ❧✐➯♥ tö❝ ✤➲✉ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠ët ✤♦↕♥ [a, b]✳ ♣ ❇➔✐ t➟♣ ❇➔✐ ✶✳ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ❣✐ỵ✐ ❤↕♥ t❤÷í♥❣ ❣➦♣ s❛✉ ✤➙② sin x =1 x→0 x ex − =1 lim x→0 x ax − = ln a lim x→0 x arc sin x =1 lim x→0 x loga (1 + x) ln(1 + x) lim = loga e ⇒ lim =1 x→0 x→0 x x (1 + x)α − =α lim x→0 x − cos x = lim x→0 x2 p x lim x = x→0 e ln x lim+ α = x→0 x ✶✳✶✳ lim ✶✳✷✳ ✶✳✸✳ ✶✳✹✳ ✶✳✺✳ ✶✳✻✳ ✶✳✼✳ ✶✳✽✳ ✶✳✾✳ ✶✳✶✵✳ lim (1 + x1 )x = lim (1 + x) x = e x→±∞ x→0 ❇➔✐ ✷✳ ❚➻♠ ❝→❝ ❣✐ỵ✐ ❤↕♥ s❛✉ x − x2 − x + ✷✳✶✳ lim x→1 x − x2 + x − ✹ ✷✳✷✳ ✷✳✸✳ ✷✳✹✳ ✷✳✺✳ x1000 − 2x + lim x→1 x500 − 2x + (1 + mx)n − (1 + nx)m lim m, n ∈ N x→0 x2 (2x − 3)20 (2x + 2)30 lim x→+∞ (2x + 1)50 √ √ √ x+ 3x+ 4x √ lim x→+∞ 2x + ✷✳✻✳ (2x − 3)20 (2x + 2)30 lim x→+∞ (2x + 1)50 ✷✳✼✳ x4 + x + √ lim x→+∞ 8x3 x + x2 + x + x2 − x→1 x2 − 4x + √ x−1 ✷✳✾✳ lim x→1 x2 − √ √ x2 + x − x2 − x) ✷✳✶✵✳ lim ( ✷✳✽✳ lim x→+∞ ✷✳✶✶✳ lim (x − √ x→+∞ ✷✳✶✷✳ lim (x − x→+∞ ✷✳✶✸✳ √ x2 − 2x) x3 − 3x2 + 4) √ √ lim ( x3 − 3x2 + 3x + − x3 − 3x2 + 4) x→+∞ ✷✳✶✹✳ √ √ lim ( x3 + 4x2 + + − x2 + x3 ) x→+∞ ❇➔✐ ✸✳ ✸✳✶✳ ✸✳✷✳ ✸✳✸✳ ✸✳✹✳ ❚➻♠ ❝→❝ ❣✐ỵ✐ ❤↕♥ s❛✉ √ cos x − cos x lim x→0 sin2 x + sin x − cos x lim x→0 + sin px − cos px sin x − sin a lim x→a x−a cos(a + 2x) − cos(a + x) + cos a lim x→0 x2 √ ✺ ✸✳✺✳ ✸✳✻✳ ✸✳✼✳ ✸✳✽✳ sin2 x + sin x − lim π sin2 x − sin x + x→ tan3 x − tan x lim π π cos(x + x→ x2 lim √ x→0 + sin x cos x √ √ lim (sin x + − sin x) x→+∞ ln cos ax , b=0 x→0 ln cos bx sin2 2x ✸✳✶✵✳ lim x→0 sin 4x sin2 2x + sin x ✸✳✶✶✳ lim x→0 sin 3x √ √ + sin x − − tan x ✸✳✶✷✳ lim x→0 x √ √ + sin x + + sin x − ✸✳✶✸✳ lim x→0 sin 2x x − sin 5x + sin2 x ✸✳✶✹✳ lim x→0 4x + arcsinx +x2 arcsin 3x − sin 5x + sin2 x ✸✳✶✺✳ lim x→0 sin x + arcsin2 x + x2 arcsin(x3 + tan2 3x) ✸✳✶✻✳ lim x→0 − cos x + sin2 x ✸✳✾✳ lim ✸✳✶✼✳ lim + sin2 x + arcsin3 x (ex − 1)2 h(cos x) + x→0 ❇➔✐ ✹✳ ❚➻♠ ❝→❝ ❣✐ỵ✐ ❤↕♥ s❛✉ ✹✳✶✳ lim x→+∞ ✹✳✷✳ lim x→+∞ ✹✳✸✳ lim x→+∞ x+1 3x + 2x 3x2 − x + 2x2 + x + x2 + x−1 x+1 x2 − x2 1−x ✻ lim ✹✳✹✳ x→+∞ ✹✳✺✳ x2 + x2 − lim + x x2 cot2 x x→0 ✹✳✻✳ ✹✳✼✳ ✹✳✽✳ ✹✳✾✳ tan x + sin13 x lim x→0 sin x + arcsin x + x1 lim x→0 sin x + x lim sin + cos x1 x→+∞ x x x2 +1 x lim 2e x + − x→0 ✹✳✶✵✳ lim+ x→0 1+e1/x ❇➔✐ ✺✳ ❳➨t t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❝→❝ ❤➔♠ sè ✺✳✶✳ f (x) = ✺✳✷✳ ✺✳✸✳ ♥➳✉ x2 2−x x=0 ♥➳✉ sin x f (x) = |x| f (x) = ✺✳✹✳ sin x | x | x=0 ♥➳✉ x=0 ♥➳✉ x>0 ♥➳✉ ♥➳✉ 0≤x≤1 1 1 f (x) = lim , x ∈ [0, +∞) n→+∞ + xn ❚➻♠ a ✤➸ ❤➔♠ sè s❛✉ ❧✐➯♥ tö❝ cos ✺✳✺✳ ❇➔✐ ✻✳ ✻✳✶✳ f (x) = 2ex a + 2x ✼ ♥➳✉ ♥➳✉ x 0) n=1 ∞ 2n − 3n ✶✳✻✳ 5n n=1 ∞ sin n √ ✶✳✼✳ n=1 n n + ∞ π ✶✳✽✳ sin n n=1 ✶✳✾✳ ∞ sin(n + 1) n n=1 + n + ✶✳✶✵✳ n2 + 3 n=1 4n + 5n ∞ ✶ ∞ + 3n ✶✳✶✶✳ 5n n=1 ∞ √ n ✶✳✶✷✳ n 4n − n=1 ∞ ✶✳✶✸✳ n=1 n(n + 3) 2n 7n (n!)2 2n n=1 n ∞ n+1 ✶✳✶✺✳ (−1)n n n=1 ∞ ✶✳✶✹✳ ∞ ✶✳✶✻✳ n=1 ✶✳✶✼✳ ∞ n=1 (2n + 1)(2n + 5) ∞ ✶✳✶✽✳ n n=1 ∞ ✶✳✶✾✳ n=1 ✶✳✷✵✳ ∞ n2 1− n 3n n sin 2n3 − 2n2 + n n+1 √ ln n n−1 n=1 ∞ ✶✳✷✶✳ n=1 ✶✳✷✷✳ n(n−1) n−1 n+1 1 √ −√ n n+2 ∞ 1.3.5 .(2n − 1) 22n (n − 1)! n=1 n2 √ n9 + n=1 n3 + ∞ ✶✳✷✹✳ n arctan n n=1 ∞ ✶✳✷✸✳ ∞ ✶✳✷✺✳ n=1 ❇➔✐ ✷✳ n n+1 n ❑❤↔♦ s→t sü ❤ë✐ tö ✤➲✉ ❝õ❛ ❝❤✉é✐ ❤➔♠ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤ ✤➣ ❝❤♦ ✷✳✶✳ ∞ n=1 3n sin x tr➯♥ [0, 1]✳ 5n ✷ (n!)3 cos x tr➯♥ (−∞, +∞)✳ ✷✳✷✳ n=1 (3n)! ∞ √ ✷✳✸✳ tr➯♥ [0, +∞)✳ n=1 n n + x ∞ sin x √ tr➯♥ R ✭①➨t t➼♥❤ ❤ë✐ tö✮✳ ✷✳✹✳ n=1 n n + ∞ ∞ sin(2n x) ✷✳✺✳ ✳ 3n n=1 ∞ ✷✳✻✳ (−1) n n n=1 sin(nx) tr➯♥ R✳ cos(nx) + √ n n+1 ∞ xn tr➯♥ [−1, 1]✳ n=1 n(+1) ∞ ✷✳✽✳ tr➯♥ [0, 1]✳ n n n=1 x ∞ xn √ tr➯♥ (−1, 1) ✭①➨t t➼♥❤ ❤ë✐ tö✮✳ ✷✳✾✳ n=1 n n + ∞ ✷✳✶✵✳ tr➯♥ [0, +∞)✳ n n n=1 + x ✷✳✼✳ ✷✳✶✶✳ ∞ (1 − x)xn ❦❤ỉ♥❣ ❤ë✐ tư ✤➲✉ tr➯♥ [0, 1]✳ n=1 ✷✳✶✷✳ ∞ xn e−nx tr➯♥ [0, +∞)✳ n=1 ❇➔✐ ✸✳ ❚➼♥❤ tê♥❣ tr➯♥ ♠✐➲♥ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤➔♠ ✸✳✶✳ S(x) = ∞ nxn+1 n=1 x2 + x ✳ ✸✳✷✳ S(x) = n x ❜✐➳t = (1 − x)3 n=1 n=1 n ∞ nx ✸✳✸✳ S(x) = (−1) ✳ n n=1 ∞ ✸✳✹✳ S(x) = ∞ ∞ n (−1) n sin n+1 3n n=1 ✸✳✺✳ S(x) = ∞ (−1) n=1 nx n+1 n (x) ✳ ✳ ✸ x2 + x ✳ ✸✳✻✳ S(x) = nx ❞ü❛ ✈➔♦ ❦➳t q✉↔ nx = 1−x n=1 n=1 ∞ ✸✳✼✳ S(x) = ∞ n n nx2 ✳ + x3 n n=1 ∞ ∞ ✸✳✽✳ S(x) = (−1)n 2nx2n ✳ n=1 ∞ ✸✳✾✳ S(x) = n3 xn ❜✐➳t n=1 ❇➔✐ ✹✳ ∞ n2 xn = n=1 x2 + x ✳ (1 − x)3 ❚➻♠ ❜→♥ ❦➼♥❤ ✈➔ ♠✐➲♥ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤➔♠ ❧ô② t❤ø❛ ✹✳✶✳ (n!)3 n x n=1 (3n)! ∞ n3 + ✹✳✷✳ (x − 1)2n n! n=1 ∞ xn √ ✹✳✸✳ n n=1 ∞ xn ✳ ✹✳✹✳ n n.4 n=1 2 n ❇➔✐ ✺✳ ❈❤♦ fn (x) = n x(1 − x ) tr➯♥ [0, 1]✳ ∞ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ fn (x) → f (x) ≡ ✈➔ 1 n→∞ 0 ❇➔✐ ✻✳ lim fn (x)dx fn (x)dx = lim n→∞ ❳➨t sü ❤ë✐ tö ✤➲✉ ❝õ❛ ❝→❝ ❞➣② ❤➔♠ tr➯♥ ❝→❝ t➟♣ ✻✳✶✳ fn (x) = xn ❛✳ ≤ x ≤ ❜✳ ≤ x ≤ ✻✳✷✳ fn (x) = xn − xn−1 , x ∈ [0, 1] ✻✳✸✳ fn (x) = xn − x2n , x ∈ [0, 1] ✻✳✹✳ fn (x) = nx , x ∈ [0, 1] 1+n+x ✹ ✻✳✺✳ fn (x) = x2 + ✻✳✻✳ fn (x) = n , x ∈ [0, 1] n2 x+ √ − x , x ∈ (0, +∞) n sin(nx) , x∈R n x ✻✳✽✳ fn (x) sin , x ∈ R n ✻✳✼✳ fn (x) = ✻✳✾✳ fn (x) = arctan(nx), x ∈ (0, +∞) ✻✳✶✵✳ fn (x) = x arctan(nx), x ∈ (0, +∞)✳ ❇➔✐ ✼✳ ❳➨t sü ❤ë✐ tö ✤➲✉ ❝õ❛ ❝→❝ ❝❤✉é✐ ❤➔♠ tr➯♥ ❝→❝ t➟♣ x2n ✼✳✶✳ ❈❤✉é✐ (−1) ❝â ❤ë✐ tö ✤➲✉ tr➯♥ R ❦❤æ♥❣❄ (2n)! n=1 ∞ √ √ (1 − x2 )n ❝â ❤ë✐ tö ✤➲✉ tr➯♥ (− 2, 2) ❦❤æ♥❣❄ ✼✳✷✳ ❈❤✉é✐ x n n=1 ❇➔✐ ✽✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② ❤➔♠ fn (x) = ❤ë✐ tö ✤✐➸♠ ✤➳♥ f (x) ≡ nx + tr➯♥ (0, 1) ♥❤÷♥❣ ❦❤ỉ♥❣ ❤ë✐ tư ✤➲✉ tr➯♥ ❦❤♦↔♥❣ ✤â✳ ∞ n x2 ❇➔✐ ✾✯✳ ❈❤♦ ❞➣② ❤➔♠ (fn (x)) = ( ), ≤ x ≤ 1, n = 1, 2, x + (1 − nx)2 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ fn (x) ❜à ❝❤➦♥ ✤➲✉ tr➯♥ [0, 1], lim fn (x) = ♥❤÷♥❣ ❦❤ỉ♥❣ n→∞ ❝â ♠ët ❞➣② ❝♦♥ ♥➔♦ ❤ë✐ tö ✤➲✉ tr➯♥ [0, 1]✳ x ♥➳✉ ≤ x ≤ − x ♥➳✉ ≤ x ≤ ✈➔ φ(x + 2) = φ(x) ✈ỵ✐ x ∈ / [0, 2]✳ ∞ n φ(4n x)✱ ❦❤✐ ✤â f ❧✐➯♥ tö❝ tr➯♥ R ♥❤÷♥❣ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ∃f (x) = n=0 ❇➔✐ ✶✵✯✳ ❈❤♦ φ(x) = ❦❤æ♥❣ ❦❤↔ ✈✐ t↕✐ ♠å✐ ✤✐➸♠✳ ❇➔✐ ✶✶✳ ❈❤♦ ❝❤✉é✐ ❤➔♠ ∞ n=1 + xn ✶✶✳✶✳ ❚➻♠ ♠✐➲♥ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ✺ ✶✶✳✷✳ ❳➨t sü ❧✐➯♥ tö❝ ❝õ❛ tê♥❣ ❝❤✉é✐ ❤➔♠✳ ❇➔✐ ✶✷✳ ❇➔✐ ✶✸✳ ❇➔✐ ✶✹✳ ❇➔✐ ✶✺✳ nx2 tr➯♥ [0, +∞)✳ 3 n=1 x + n ∞ ❚➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ f (x) = ✳ 2 n=1 n + x ∞ ❚➻♠ sè t❤ü❝ x s❛♦ ❝❤♦ xn = ✳ 11 n=1 ❳➨t t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ f (x) = ❳➨t t➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ t↕✐ x = −3 ✈➔ ♣❤➙♥ ❦ý t↕✐ x = 5✳ ❇➔✐ ✶✻✳ ∞ ∞ an 2n ❜✐➳t ❝❤✉é✐ ❤➔♠ n=1 ❇➔✐ ✶✼✳ an xn ❤ë✐ tö n=1 ❳➨t t➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤➔♠ ❤ë✐ tö t↕✐ x = −2 ✈➔ ♣❤➙♥ ❦ý t↕✐ x = 4✳ ∞ ∞ an 3n ❜✐➳t ❝❤✉é✐ ❤➔♠ n=1 ∞ an x n n=1 ❚➻♠ ❜→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤➔♠ ∞ n!3n n x ✶✼✳✶✳ n n=1 n ✶✼✳✷✳ ∞ n! n x 2n n=1 n (n!)3 n x ✶✼✳✸✳ n=1 (3n)! ∞ ✶✼✳✹✳ ∞ n! n x n n=1 n x2n ✶✼✳✺✳ 2n n=1 ∞ ∞ nn+3 x2n ✶✼✳✻✳ n n=1 (4n + 1) ✶✼✳✼✳ ❇➔✐ ✶✽✳ n3 + (x − 1)2n ✳ n! n=1 ∞ ❚➻♠ ❜→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ∞ ∞ n=0 an xn ❧➔ 10✳ n=0 ✻ 2n(n − 1)an xn−2 ❜✐➳t ❜→♥ ❦➼♥❤ ∞ sin(nx), x ∈ [−π, π]✱ ❝→❝ ❤➺ sè ♥➔♦ tr♦♥❣ ❦❤❛✐ n=1 n tr✐➸♥ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ sè f (x) tr➯♥ ✤♦↕♥ [−π, π] ❜➡♥❣ ✵❄ ❇➔✐ ✶✾✳ ❇➔✐ ✷✵✳ ❈❤♦ f (x) = ❑❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r ❝→❝ ❤➔♠ sè s❛✉✱ ❜✐➳t ❝❤ó♥❣ ❧➔ ♥❤ú♥❣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦ý T = 2π ✳ 1, −π ≤ x < 2, ≤ x ≤ π sin(2t), ≤ t < π s(t) = 0, π ≤ x ≤ 2π 2π + x, −π ≤ x < f (x) = 2, 0≤x≤π π, −π ≤ x < f (x) = π − x, ≤ x ≤ π −1, −π ≤ x < π 0, ≤ x ≤ f (x) = π 1,