TRABAJO COLABORATIVO UNO ECUACIÓN DIFERENCIAL

TRABAJO COLABORATIVO UNO

MARLY JOHANA FIESCO RIVERA

GRUPO: 10410 – 88

TUTOR:

HAROLD PEREZ

UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA - UNAD

PROGRAMA CÁLCULO DIFERENCIAL

TARQUI - HUILA

MARZO 2011

FASE 1

Hallar los 5 primeros términos de las siguientes sucesiones:

Un=(1/(3ⁿ^(+1) ) )ɳ≥1 = ( 1/9 ,1/(27 ) ,1/(81 ) ,1/243 ,1/729 … )

ɳ = 1 1/(3¹⁺¹) = 1/9

ɳ = 2 1/(3 ²⁺¹) = 1/3³ = 1/27

ɳ = 3 1/(3³⁺¹) =1/(3⁴) = 1/81

ɳ = 4 1/(3⁴⁺¹)= 1/(3⁵)= 1/243

ɳ = 5 1/(3⁵⁺¹)= 1/(3⁶)= 1/729

Un = ( 3/(3ⁿ^(-4) ) ) ɳ≥1=( -1, 1.5,0.20,0.125,0.16… )

ɳ = 1 3/(3( 1 )– 4 )= 3/(3-4) = -1/1= - 1

ɳ = 2 3/(3(2)- 4)= 3/(6 -4)= 3/2=1.5

ɳ = 3 1/(3 ( 3 )- 4)= 1/(9 -4)= 1/5=0.20

ɳ = 4 1/(3 ( 4 )- 4)= 1/(12 –4 )= 1/8=0.125

ɳ = 5 1/(2 ( 5 )– 4 )= 1/(30 -4)= 1/6=0.16

Wɳ = ( 1/(n -1) )ⁿ ɳ ≥2= ( 1,0.5,0.33,0.25,0.20… )

ɳ = 1 1/(2 -1)= 1/1=1

ɳ = 2 1/(3 –1 )= 1/2= 0.5

ɳ = 3 1/(4 -1)= 1/(3 ) = 0.33

ɳ = 4 1/(5 –1 )= 1/4 =0.25

ɳ = 5 1/(6 –1 )= 1/5 = 0.20

Identificar el término general dados el primer término y la relación de recurrencia.

U0= 2; Un= Un – 1 +1

U1= U0 +1=2 +1=3

U2 = U 2-1 +1= U1 +1=3 +1=4

U3= U 3 – 1 +2= U2 +1=4 +1=5

Uɳ= ɳ +2

U0=4; Uɳ = ( Uɳ -1)/5

U1= (U1 -1)/5= U0/5¹ = 4/5¹

U2 = ( U2 -1 )/5= ( U1 )/5=4 ( 1/5^2 )

U3 = (U3 -1)/5= U2/5=4 ( 1/5^3 )

U4= ( U4 -1 )/5= ( U3)/5 =4 ( 1/5^4 )

Uɳ=4 ( 1/(5 ) )ⁿ

ɳ ≥0

Wɳ = ( 2/(1 〖^-〗ɳ) ) ɳ≥2

Uɳ + 1 > Uɳ

Uɳ + 1 - Uɳ > 0

2/(1 - ( ɳ +1 ) ) ≥ 2/(1 - ɳ)

2/(1 - ɳ -1 ) ≥ 2/(1 - ɳ)

-2/ɳ ≥ 2/(1 - ɳ)

2/ɳ ≤ 2/(ɳ -1)

2 (ɳ - 1) ≤ 2 ɳ

2ɳ - 2 ≤ 2ɳ

2ɳ - 2ɳ ≤ 2

0 ≤ 2

Xɳ = 2 ⁻ⁿ

Xɳ = 1/(2 ⁿ)

Uɳ + 1 ˂ Uɳ

1/(2ⁿ ⁺¹) ˂ 1/2ⁿ

2 ⁿ ˂ 2ⁿ ⁺ ¹

Vɳ = ( (2ɳ +1)/ɳ )ɳ≥1

M = 1

Uɳ ≤ M

(2ɳ+1)/ɳ - 1

(2ɳ+1-ɳ)/ɳ= (ɳ+1)/ɳ

ɳ+1 ≤ 0 Y ɳ > 0

(ɳ +1)/ɳ ≤1

Para ɳ=0

FASE DOS

Vɳ = ( (2ɳ +1)/ɳ )ɳ≥1

Para M = 1 es una cota superior

Para m = -1 es una cota inferior

Vɳ = 3 + 2 (ɳ-1)

Vɳ = 3 + 2ɳ - 2 = 2ɳ + 1

Vɳ = 2ɳ+1

ɳ = 1

Vɳ = 2 (1) + 1 = 3

El primer término es 3

Uɳ + 1 = U+ r

r = 1 Diferencia común

ɳ = 0

U0= 1

U₁+1= U₁+1 = 2+1 = 3

U₂+1 = U₂+1 = 3 + 1 = 4

Uɳ+ 1 = Uɳ- 4

Uɳ = Uₐ+(ɳ-a)r

Uɳ = U₁+(ɳ-1)r

Uɳ = 3 +(ɳ-1)(-4)

Uɳ = 3 - 4ɳ+4

Uɳ = 7 - 4ɳ

U₁ = 7 – 4 (1) = 7-4 = 3

U₂ = 7 – 4 (2) = 7-8 = -1

U₃= 7-4 (3) = 7-12 = -5

U₄= 7-4(4) =7-16 = -9

U₅= 7-4(5) = 7-20 = -13

U₆= 7-4 (6) = 7-24 = -17

U₇= 7-4 (7) = 7-28 = -21

U₁= 1

Uɳ= 15

15= 200

5 = (ɳ(ɳ+1))/2

200= (ɳ(ɳ+1))/2

ɳ(ɳ+1) = 400

ɳ= 400

ɳ²-ɳ-400 = 0

x=(-(-1)±√(〖(-1)〗^2-4(1)(400)))/(2(1))

x=(1±√(1+1600))/2

X= 41.01/2=20,50

X = 99.01/2=19,50

X= 20

r= (Uɳ- Uₐ)/(ɳ-a)

r= (15-1)/(20-1)= 14/19

r= 14/19

progresión geométricas

U₁= qⁿ. Uₒ

Uɳ= qⁿ- ͣ.Uₐ

S= (qⁿ-1)/(q-1)

Los primeros 60 números naturales:

{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60}

Los múltiplos de 5 ≤ 180

{5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,110,115,120,125,130,135,140,145,150,155,160,165,170,175,180…}

Los 10 primeros múltiplos de 9:

{9, 18, 27, 36, 45, 54, 63, 72, 81,90…}

FASE TRES

Uɳ= (1/3)ⁿɳ≥1

ɳ =1 ( 1/3) ¹= 1/3

ɳ = 2 (1/3^2 )= 1/9

ɳ = 3 (1/3^3 )= 1/27

ɳ = 4 (1/(3⁴))= 1/81

ɳ = 5 (1/3^5 )= 1/243

Vɳ= (2ⁿ/(2^( ⁿ^(+2)) ))

Converge a 0

ɳ= 1 2¹/2³= 1/2²= 1/4

ɳ= 2 2²/2³= 1/2¹= 1/2

ɳ= 10 (2¹°)/2¹²= 1/2²= 1/4

ɳ= 100 (2¹°°)/(2¹°²)= 1/2²= 1/4

Converge a 0

Sɳ= {(3-ɳ)/(2+3ɳ)}

Tiene límite -1/3

Lim.

ɳ⟞∞ (3-ɳ)/(2+3ɳ)

Lim.

ɳ⟞∞ (3/ɳ- ɳ/ɳ)/(2/ɳ+ 3ɳ/ɳ)= (0-1)/(0+3)= -1/3

Wɳ= {3/7 (2) ⁿ^(-1)}

Lim.

X⟞∞ {3/7 (2) ⁿ^(-1)}

Lim.

X⟞∞ {3/7 2ⁿ/2¹}

Lim.

X⟞∞ ({3/7 2ⁿ/2¹})/( 2ⁿ/2¹)

Lim.

X⟞∞ {(3/7 (1))/0}

Divergente

Wɳ= { √ɳ²- 2ɳ - ɳ

⎸√ɳ²- 2ɳ - ɳ - (-1) ⎸<ε

⎸√ɳ²- 2ɳ - ɳ +1 ⎸<ε

⎸√ɳ²- 2ɳ - ɳ +1= ⎸√ɳ²- 2ɳ - ɳ +1

√ɳ²- 2ɳ - ɳ +1 <ε

⎸√ɳ²- 2ɳ - ɳ +1 ⎸≤ε

Lo que significa que converge a -1

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