Series Aritmetico-geometricas PDF

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Ejercicios resueltos...

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Cálculo Infinitesimal

Suma de series

S eries Aritm e tico – Geome tricas ∑ 𝑃𝑜𝑙(𝑛) ∙ 𝑔𝑒𝑜𝑚é𝑡𝑟𝑖𝑐𝑎 ↑ 𝑎𝑟𝑖𝑡𝑚é𝑡𝑖𝑐𝑎 𝑑𝑒 𝑜𝑟𝑑𝑒𝑛 𝑑𝑒𝑙 𝑔𝑟𝑎𝑑𝑜 𝑑𝑒𝑙 𝑝𝑜𝑙𝑖𝑛𝑜𝑚𝑖𝑜 ( 𝑜𝑟𝑑𝑒𝑛 k) 𝑆 = ⋯ … … … … .. − 𝑟 ∙ 𝑆 = ⋯………… 𝑟 𝑒𝑠 𝑙𝑎 𝑟𝑎𝑧ó𝑛 𝑑𝑒 𝑙𝑎 𝑔𝑒𝑜𝑚é𝑡𝑟𝑖𝑐𝑎 𝑆 − 𝑟 ∙ 𝑆 = ⋯ 𝑠𝑒 𝑜𝑏𝑡𝑖𝑒𝑛𝑒 𝑜𝑡𝑟𝑎 𝑎𝑟𝑖𝑡𝑚é𝑡𝑖𝑐𝑜 − 𝑔𝑒𝑜𝑚é𝑡𝑟𝑖𝑐𝑎 𝑝𝑒𝑟𝑜 𝑙𝑎 𝑎𝑟𝑖𝑡𝑚é𝑡𝑖𝑐𝑎 𝑒𝑠 𝑑𝑒 𝑢𝑛 𝑜𝑟𝑑𝑒𝑛 𝑚𝑒𝑛𝑜𝑠. 𝑆𝑒 𝑟𝑒𝑝𝑖𝑡𝑒 ℎ𝑎𝑠𝑡𝑎 𝑙𝑙𝑒𝑔𝑎𝑟 𝑎 𝑢𝑛𝑎 𝑔𝑒𝑜𝑚é𝑡𝑟𝑖𝑐𝑎. ∞

(𝑎 + 𝑏) ∙ 𝑟 − 𝑏 ∙ 𝑟 2 ∑(𝑎 ∙ 𝑛 + 𝑏) ∙ 𝑟 = (1 − 𝑟)2 𝑛

𝑛=1

Ejercicio 1: 𝑺𝒖𝒎𝒂𝒓:

∞

𝟑 𝟕 𝟏𝟏 𝟏𝟓 𝟏 + + + + ⋯ = ∑(𝟒𝒏 − 𝟏) ∙ ( )𝒏 𝟐 𝟐 𝟒 𝟖 𝟏𝟔 𝒏=𝟏

1

1

1

1

1

𝑆 = 3 ∙ ( 2) + 7 ∙ (2)2 + 11 ∙ (2)3 + 15 ∙ (2)4 + ⋯+(4𝑛 − 1) ∙ ∙ (2)𝑛 1

1

1

1

1

1

− 2 𝑆 = −3 ∙ (2)2 + 7 ∙ (2)3 + 11 ∙ (2)4 + 15 ∙ (2)5 + ⋯+(4𝑛 − 1) ∙ (2)𝑛+1 1 3 1 1 1 1 1 1 𝑆 = + 4 ∙ [( )2 + ( )3 + ( )4 + 15 ∙ ( )5 + ⋯ + ( )𝑛 ] − (4𝑛 − 1) ∙ ( )𝑛+1 2 2 2 2 2 2 2 2 Progresión geométrica 3 1 𝑆= +4∙ 2 2 1 3 𝑆= +2 2 2

1 4

1−

1 2

−0

𝑺=𝟕

Academia San Claudio

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Página 1

Cálculo Infinitesimal

Suma de series

Ejercicio 2: 𝑺𝒖𝒎𝒂𝒓: 𝟓 +

∞

𝟖 𝟏𝟏 𝟏𝟒 𝟏 + + + ⋯ = ∑(𝟑𝒏 + 𝟐) ∙ ( )𝒏−𝟏 𝟐 𝟐 𝟒 𝟖 𝒏=𝟏

1

1

1

1

1

1

1

1

𝑆 = 5 + 8 ∙ ( 2) + 11 ∙ ( 2)2 + 14 ∙ (2)3 + 17 ∙ (2)4 + ⋯+(3𝑛 + 2) ∙ (2)𝑛−1 1

1

1

1

− 𝑆 = −5 ∙ ( ) − 8 ∙ ( )2 − 11 ∙ ( )3 − 14 ∙ ( )4 − 17 ∙ ( )5 + ⋯+(3𝑛 + 2) ∙ (2)𝑛 2 2 2 2 2 2 1 1 1 1 1 1 1 1 𝑆 = 5 + 3 ∙ [( ) + ( )2 + ( )3 + ( )4 + ( )5 + ⋯ + ( )𝑛 ] − (3𝑛 + 2) ∙ ( )𝑛 2 2 2 2 2 2 2 2 Progresión geométrica 1 1 𝑆= 5+3∙ 2 −0 1 2 1− 2 1 𝑆 = 5+3 2

𝑺 = 𝟏𝟔

Ejercicio 3: ∞

𝟕 𝟗 𝟏 𝟓 + + ⋯ = ∑(𝟐𝒏 + 𝟑) ∙ ( )𝒏 𝑺𝒖𝒎𝒂𝒓: + 𝟖 𝟔𝟒 𝟓𝟏𝟐 𝟖 𝒏=𝟏

1

1

1

1

1

𝑆 = 5 ∙ ( ) + 7 ∙ ( )2 + 9 ∙ ( )3 + 11 ∙ ( )4 + ⋯+(2𝑛 + 3) ∙ (8)𝑛 8 8 8 8 1

1

1

1

1

1

− 𝑆 = −5 ∙ ( )2 − 7 ∙ ( )3 − 9 ∙ ( )4 − 11 ∙ ( )5 + ⋯+(2𝑛 + 3) ∙ ( )𝑛+1 8 8 8 8 8 8

7 5 1 1 1 1 1 1 𝑆 = + 2 ∙ [( )2 + ( )3 + ( )4 + ( )5 + ⋯ + ( )𝑛 ] − (2𝑛 + 3) ∙ ( )𝑛+1 8 8 8 8 8 8 8 8 Progresión geométrica 1 ( )2 5 7 𝑆= +2∙ 8 −0 1 8 8 1−8

7 5 1 35 + 2 37 𝑆= + = = 8 8 28 56 56

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𝑺=

𝟑𝟕 𝟒𝟗

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Página 2

Cálculo Infinitesimal

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Ejercicio 4: ∞

𝟏𝟎 𝟐𝟏 𝟑𝟖 𝟏 𝑺𝒖𝒎𝒂𝒓: + + + ⋯ = ∑(𝟑𝒏𝟐 + 𝟐𝒏 + 𝟓) ∙ ( )𝒏 𝟖 𝟖 𝟔𝟒 𝟓𝟏𝟐 𝒏=𝟏

1

1

1

1

1

𝑆 = 10 ∙ ( ) + 21 ∙ ( )2 + 38 ∙ ( )3 + 61 ∙ ( )4 + ⋯+(3𝑛2 + 2𝑛 + 5) ∙ (8)𝑛 8 8 8 8 1

1

1

1

1

1

− 𝑆 = −10 ∙ ( )2 − 21 ∙ ( )3 − 38 ∙ ( )4 − 61 ∙ ( )5 + ⋯+(3𝑛 2 + 2𝑛 + 5) ∙ (8)𝑛+1 8 8 8 8 8 10 1 1 1 1 1 7 𝑆= + 11 ∙ ( )2 + 17 ∙ ( )3 + 23 ∙ ( )4 + 29 ∙ ( )5 + ⋯ − (3𝑛 2 + 2𝑛 + 5) ∙ ( )𝑛+1 8 8 8 8 8 8 8 Progresión aritmético-geométrica 11 + 17 + 23 + 29 + ⋯ 𝑑 = 6 𝑎1 = 11

𝑺𝒖𝒎𝒂𝒓:

𝑎𝑛 = 11 + (𝑛 − 1) ∙ 6 = 6𝑛 + 5

∞

𝟏𝟏 𝟏𝟕 𝟐𝟑 𝟏 + + + ⋯ = ∑(𝟔𝒏 + 𝟓) ∙ ( )𝒏+𝟏 𝟖 𝟔𝟒 𝟓𝟏𝟐 𝟒𝟎𝟗𝟔 𝒏=𝟏

1

1

1

1

𝑆´ = 11 ∙ ( )2 + 17 ∙ ( )3 + 23 ∙ ( )4 + ⋯+(6𝑛 + 5) ∙ (8)𝑛+1 8 8 8 1

1

1

1

1

− 8 𝑆´ = −11 ∙ (8)3 − 17 ∙ (8)4 − 23 ∙ (8)5 + ⋯+(6𝑛 + 5) ∙ (8)𝑛+2 7 11 1 1 1 1 1 𝑆´ = + 6 ∙ [( )3 + ( )4 + ( )5 + ⋯ + ( )𝑛+1 ] − (6𝑛 + 5) ∙ ( )𝑛+2 8 64 8 8 8 8 8 Progresión geométrica 1 (8)3 7 11 −0 +6∙ 𝑆´ = 1 64 8 1− 8

7 11 1 83 𝑆´ = +6∙ = 8 64 448 448

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𝑺´ =

𝟖𝟑 𝟑𝟗𝟐

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Página 3

Cálculo Infinitesimal

Suma de series

10 1 1 1 1 1 7 𝑆= + 11 ∙ ( )2 + 17 ∙ ( )3 + 23 ∙ ( )4 + 29 ∙ ( )5 + ⋯ − (3𝑛 2 + 2𝑛 + 5) ∙ ( )𝑛+1 8 8 8 8 8 8 8 10 𝟖𝟑 7 𝑆= + −0 8 8 𝟑𝟗𝟐

𝟓𝟕𝟑 𝟏𝟎 𝟖𝟑 + 𝟑𝟗𝟐 𝟓𝟕𝟑 𝟖 = 𝟑𝟗𝟐 = 𝑺= 𝟕 𝟕 𝟑𝟒𝟑 𝟖 𝟖

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