terça-feira, 24 de março de 2020

Raíz, potência perfeita e Números espelhos




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Números espelhos III
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UNB-Matemática
Autor: Luiz Augusto Prado 1807787/DF
Blog: tivideotutoriais.blogspot.com

Até o momento vimos:
- Raíz Perfeita
- Potência Perfeita
- Algumas propriedades dos números espelhos.

Vamos continuar estudando e apresentando novas propriedades dos números espelhos.

Números espelhos seguem um padrão. Para todos os números espelhos existe um n que satisfaça:
a = [n/(n+1)]n+1 que está abaixo de 1/e
b = [n/(n+1)]n está entre 1/e e 1

∀ n ∈ ℝ+
Se n → ∞+ então
a → 1/e pela esquerda
b → 1/e pela direita
aa = bb → 1/e1/e pela direita

Exemplo: Se n=100000
→ a = [n/(n+1)]n+1= 0.3678776017869739 → aa = 0.6922006275585294 > 0.6922006275553464
→ b = [n/(n+1) ] n = 0.36788128056299174 → bb = 0.6922006275585294 > 0.6922006275553464

∀ n ∈ ℝ+
Se n → 0+ então
a → 0+ pela direita
b → 1- pela equerda
aa = bb → 1- pela esquerda

Exemplo: Se n=0.0001
→ a = [n/(n+1)]n+1= 0.0000998979482045963 → aa = 0.9990802271454703 < 1
→ b = [n/(n+1) ] n = 0.9990793799941674 → bb = 0.9990802271454703 < 1

b/a= ln(a)/ln(b) = (n+1)/n = n [ ln(n+1) - ln(n) ] / ln(n)
a/b= ln(b)/ln(a) = n/(n+1)= n-[ ln(n+1) - ln(n) ] / ln(n)

Note que se n=1 não existirá potência que o eleve que dê resultado diferente de 1. Portanto não podemos utilizar formulas onde ln(1)=0 façam divisão.

Se k = 1/n + 1 , também podemos ter as equivalencias abaixo:
b/a+a/b= (2n2+2n+1)/(n2+n) = ([1/n + 1]2+1) / (1/n + 1)= (k2+1) / k
b/a-a/b= (2n+1)/(n2+n) = ([1/n + 1]2 - 1) / (1/n + 1)= (k2 - 1) / k

Se a/b= n/(n+1)
a = [ n/(n+1)]n+1= [a/b]n+1
b = [ n/(n+1) ] n = [a/b]n

Se temos:
a = [n/(n+1)]n+1
b = [n/(n+1)]n
E se existe um z que satisfaça as equivalências:
b = z*a → z=b/a = (n+1)/n
b = a1/z → ln(b)=ln(a)/z
z = b/a = ln(a)/ln(b) = (n+1)/n
Então podemos dizer que:
b = [n/(n+1)]n = z*[n/(n+1)]n+1
b = [n/(n+1)]n = [n/(n+1)](n+1)/z

Encontrar o n de qualquer número espelho

Supondo que a ou b seja dado, e sabemos ser um espelho, se não tivermos o eu par, basta realizar o√(s') para obter os dois espelhos a e b. Podemos isolar n sabendo que:
b/a= (n+1)/n
a/b= n/(n+1)
Supondo que:
b/a = x
a/b = y
então:
(n+1)/n = x → (n+1)=n*x → x*n-n=1 → (x-1)*n=1 → n=1/(x-1)
n/(n+1) = y → n=y*(n+1) → n=y*n+y → (y-1)*n=-y → n=-y/(y-1)
Portanto:
n = 1/([b/a]-1) ou
n =-[a/b]/([a/b]-1)

Supondo que a=0.214071977364008 e b=0.5470728310413537
→ b/a = x = 23/9
→ a/b = y = 9/23
(n+1)/n = 23/9 → n=1/([23/9]-1) → n=1/([23/9]-1) → n=9/14
n/(n+1) = 9/23 → n= [-9/23]/(-14/23) → n=9/14

Encontrando j tal que b/a=nj:

Lembrando que b/a=(n+1)/n veja alguns polinômios:

x2 -x -1 = 0 tem raiz = 1.618033988749895
x3 -x -1 = 0 tem raiz = 1.324717957244746
x4 -x -1 = 0 tem raiz = 1.2207440846057596
x5 -x -1 = 0 tem raiz = 1.1673039782614187

Se n=1.618033988749895, b/a=n
Se n=1.324717957244746, b/a=n2
Se n=1.220744084605759, b/a=n3
Se n=1.167303978261418, b/a=n4

Se para qualquer b/a = (n+1)/n = nj Qual seria o j para qualquer n?
Solução:
Se b/a=nj → j*ln(n)=ln(b/a) → j=ln(b/a)/ln(n)
Se (n+1)/n=nj → j*ln(n)=ln([n+1]/n) → j=[ln(n+1)-ln(n)]/ln(n)

Supondo que n=2 encontre b/a e j tal que b/a=2j
Solução:
Se j=[ln(n+1)-ln(n)]/ln(n) → j=[ln(2+1)-ln(2)]/ln(2) → j=0.5849625007211561
Se b/a=nj → b/a=20.5849625007211561 → b/a=1.5




APÊNDICE: Lista de alguns resultados interessantes:

Raízes de alguns polinômios na forma xn -x -1 = 0
Encontrando j tal que b/a=nj:
Para n de 1 até 6
Para 1/n com n de 2 até 6
Para vários n relacionados à Pi
Para vários n relacionados à e
Para vários n relacionados com o número aureo (x tal que x2=x+1)
Para vários n relacionados com x tal que x3=x+1
Para vários n relacionados com k tal que kk=e
Alguns b/a+a/b
Alguns b/a-a/b

Raízes de alguns polinômios na forma xn -x -1 = 0

É interessante notar que uma raíz de x5-x4-1 = 0 é igual a raiz de x3-x-1 = 0

Se n = 2 : x2-x-1 = 0     → tem raiz = 1.618033988749895 →     1/(x-1) = 1.6180339887498947
Se n = 3 : x3-x-1 = 0     → tem raiz = 1.3247179572447458 →     1/(x-1) = 3.0795956234914406
Se n = 4 : x4-x-1 = 0     → tem raiz = 1.2207440846057596 →     1/(x-1) = 4.530132718101876
Se n = 5 : x5-x-1 = 0     → tem raiz = 1.1673039782614187 →     1/(x-1) = 5.977144180262483
Se n = 6 : x6-x-1 = 0     → tem raiz = 1.1347241384015194 →     1/(x-1) = 7.4225748397046125
Se n = 7 : x7-x-1 = 0     → tem raiz = 1.1127756842787055 →     1/(x-1) = 8.86715967538422
Se n = 8 : x8-x-1 = 0     → tem raiz = 1.0969815577985598 →     1/(x-1) = 10.31123878291477
Se n = 9 : x9-x-1 = 0     → tem raiz = 1.0850702454914507 →     1/(x-1) = 11.754991351241568
Se n = 10 : x10-x-1 = 0     → tem raiz = 1.0757660660868371 →     1/(x-1) = 13.198520810805705
Se n = 11 : x11-x-1 = 0     → tem raiz = 1.0682971889208412 →     1/(x-1) = 14.641891061710224
Se n = 12 : x12-x-1 = 0     → tem raiz = 1.062169167864255 →     1/(x-1) = 16.085143719206222

Encontrando os j tal que b/a=nj:

x=0.2360679774997897 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 0.10557024685139227 → b/a=1.1909830056250525
x=0.38196601125010515 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 0.18977560319749973 → b/a=1.276393202250021
x=0.6180339887498949 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 0.3361379690922773 → b/a=1.381966011250105
x=1.0000000000000002 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 0.5849625007211563 → b/a=1.5
x=1.6180339887498953 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 1.0000000000000004 → b/a=1.618033988749895
x=2.618033988749896 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 1.6828657492940997 → b/a=1.723606797749979
x=3.0795956234914392 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 2.0000000000000004 → b/a=1.754877666246693
x=4.530132718101875 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 2.9999999999999987 → b/a=1.8191725133961643
x=5.977144180262484 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 3.999999999999999 → b/a=1.856674883854503
x=7.422574839704613 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 5.000000000000003 → b/a=1.8812714616335697
x=8.867159675384224 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 5.9999999999999964 → b/a=1.8986537126286993
x=10.311238782914769 → n=1+1/x → j=[ ln(n+1)-ln(n) ] / ln(n) = 7.000000000000004 → b/a=1.9115923534820551


Se n = 2.23606797749979 (raiz de 5):

n = 2.23606797749979
a = [n/(n+1)]n+1 = 0.30234687528635806
b = [n/(n+1)]n = 0.4375605084713476
aa= bb = 0.6965181799966349
b/a = (n+1)/n = ln(a)/ln(b) = 1.4472135954999579
a/b = n/(n+1) = ln(b)/ln(a) = 0.6909830056250525
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.1381966011250104
b/a-a/b= (2n+1)/(n2+n) = 0.7562305898749053


Para n=aureo: Mesmo n onde n2=n+1:


Se n = 1.618033988749895 (Número Aureo):

n = 1.618033988749895
a = [n/(n+1)]n+1 = 0.2837025599424474
b = [n/(n+1)]n = 0.4590403846822343
aa= bb = 0.699480971720873
b/a = (n+1)/n = ln(a)/ln(b) = 1.618033988749895
a/b = n/(n+1) = ln(b)/ln(a) = 0.6180339887498949
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.23606797749979
b/a-a/b= (2n+1)/(n2+n) = 1


Para n=1.3247179572447463: Mesmo n onde n3=n+1:


Se n = 1.3247179572447463 :

n = 1.3247179572447463
a = [n/(n+1)]n+1 = 0.2705172532293568
b = [n/(n+1)]n = 0.4747246860265993
aa= bb = 0.7020999508903193
b/a = (n+1)/n = ln(a)/ln(b) = 1.7548776662466927
a/b = n/(n+1) = ln(b)/ln(a) = 0.5698402909980533
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.3247179572447463
b/a-a/b= (2n+1)/(n2+n) = 1.1850373752486394
n = 2√b / 2√a = 1.3247179572447463
n2= b/a = (n+1)/n = 1.7548776662466927
1/n2= a/b = n/(n+1) = 0.5698402909980533
2√b/n= 2√a = 0.5201127312702092
n*2√a= 2√b = 0.689002674905257
n2 + 1/n2= b/a+a/b = (2n2+2n+1)/(n2+n) = 2.3247179572447463
n2 - 1/n2= b/a-a/b = (2n+1)/(n2+n) = 1.1850373752486394


Para n=1.2207440846057596: Mesmo n onde n4=n+1:


Se n = 1.2207440846057596 :

n = 1.2207440846057596
a = [n/(n+1)]n+1 = 0.264780870793223
b = [n/(n+1)]n = 0.48168208222013253
aa= bb = 0.7033822636267789
b/a = (n+1)/n = ln(a)/ln(b) = 1.8191725133961645
a/b = n/(n+1) = ln(b)/ln(a) = 0.5497004779019703
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.368872991298135
b/a-a/b= (2n+1)/(n2+n) = 1.2694720354941942
n = 3√b / 3√a = 1.2207440846057596
n3= b/a = (n+1)/n = 1.8191725133961645
1/n3= a/b = n/(n+1) = 0.5497004779019703
3√b/n= 3√a = 0.6421387358103625
n*3√a= 3√b = 0.7838870632367206
n3 + 1/n3= b/a+a/b = (2n2+2n+1)/(n2+n) = 2.368872991298135
n3 - 1/n3= b/a-a/b = (2n+1)/(n2+n) = 1.2694720354941942


Usando valores de algumas tangentes


Se n = 1 1:

n = 1
a = [n/(n+1)]n+1 = 0.25
b = [n/(n+1)]n = 0.5
aa= bb = 0.7071067811865476
b/a = (n+1)/n = ln(a)/ln(b) = 2
a/b = n/(n+1) = ln(b)/ln(a) = 0.5
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.5
b/a-a/b= (2n+1)/(n2+n) = 1.5


Se n = 0.4142135623730951 1/(√2+1):

n = 0.4142135623730951
a = [n/(n+1)]n+1 = 0.17612178693664823
b = [n/(n+1)]n = 0.601317393588489
aa= bb = 0.7364974585056316
b/a = (n+1)/n = ln(a)/ln(b) = 3.414213562373095
a/b = n/(n+1) = ln(b)/ln(a) = 0.2928932188134525
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.7071067811865475
b/a-a/b= (2n+1)/(n2+n) = 3.1213203435596424


Se n = 0.2679491924311227 1/(√3+2):

n = 0.2679491924311227
a = [n/(n+1)]n+1 = 0.13933866964726338
b = [n/(n+1)]n = 0.6593576642299059
aa= bb = 0.7598643749205355
b/a = (n+1)/n = ln(a)/ln(b) = 4.732050807568878
a/b = n/(n+1) = ln(b)/ln(a) = 0.21132486540518708
b/a+a/b= (2n2+2n+1)/(n2+n) = 4.9433756729740645
b/a-a/b= (2n+1)/(n2+n) = 4.520725942163691


Se n = 0.198912367379658 tan(45/4):

n = 0.198912367379658
a = [n/(n+1)]n+1 = 0.11606404582325999
b = [n/(n+1)]n = 0.6995574070064391
aa= bb = 0.778834231749376
b/a = (n+1)/n = ln(a)/ln(b) = 6.027339492125848
a/b = n/(n+1) = ln(b)/ln(a) = 0.16591068104035053
b/a+a/b= (2n2+2n+1)/(n2+n) = 6.193250173166199
b/a-a/b= (2n+1)/(n2+n) = 5.8614288110854975

------


Se n = 1 1:

n = 1
a = [n/(n+1)]n+1 = 0.25
b = [n/(n+1)]n = 0.5
aa= bb = 0.7071067811865476
b/a = (n+1)/n = ln(a)/ln(b) = 2
a/b = n/(n+1) = ln(b)/ln(a) = 0.5
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.5
b/a-a/b= (2n+1)/(n2+n) = 1.5


Se n = 2.414213562373095 (√2+1):

n = 2.414213562373095
a = [n/(n+1)]n+1 = 0.3062736632680329
b = [n/(n+1)]n = 0.43313636839134256
aa= bb = 0.6959994892750379
b/a = (n+1)/n = ln(a)/ln(b) = 1.4142135623730951
a/b = n/(n+1) = ln(b)/ln(a) = 0.7071067811865475
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.121320343559643
b/a-a/b= (2n+1)/(n2+n) = 0.7071067811865477


Se n = 3.732050807568877 (√3+2):

n = 3.732050807568877
a = [n/(n+1)]n+1 = 0.32517473927676577
b = [n/(n+1)]n = 0.412305048064976
aa= bb = 0.6939891935148583
b/a = (n+1)/n = ln(a)/ln(b) = 1.2679491924311226
a/b = n/(n+1) = ln(b)/ln(a) = 0.788675134594813
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.0566243270259355
b/a-a/b= (2n+1)/(n2+n) = 0.4792740578363096


Se n = 5.027339492125848 1/[1.198912367379658-1]:

n = 5.027339492125848
a = [n/(n+1)]n+1 = 0.33505904806883746
b = [n/(n+1)]n = 0.4017064365321846
aa= bb = 0.6932463819597379
b/a = (n+1)/n = ln(a)/ln(b) = 1.198912367379658
a/b = n/(n+1) = ln(b)/ln(a) = 0.8340893189596494
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.0330016863393077
b/a-a/b= (2n+1)/(n2+n) = 0.36482304842000857


Alguns exemplos interessantes de n:

Para n de 1 até 6:


Se n = 1 (Um):

n = 1
a = [n/(n+1)]n+1 = 0.25
b = [n/(n+1)]n = 0.5
aa= bb = 0.7071067811865476
b/a = (n+1)/n = ln(a)/ln(b) = 2
a/b = n/(n+1) = ln(b)/ln(a) = 0.5
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.5
b/a-a/b= (2n+1)/(n2+n) = 1.5


Se n = 2 (Dois):

n = 2
a = [n/(n+1)]n+1 = 0.2962962962962962
b = [n/(n+1)]n = 0.4444444444444444
aa= bb = 0.697387945762143
b/a = (n+1)/n = ln(a)/ln(b) = 1.5
a/b = n/(n+1) = ln(b)/ln(a) = 0.6666666666666666
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.1666666666666665
b/a-a/b= (2n+1)/(n2+n) = 0.8333333333333334


Se n = 3 (Três):

n = 3
a = [n/(n+1)]n+1 = 0.31640625
b = [n/(n+1)]n = 0.421875
aa= bb = 0.6948233607279168
b/a = (n+1)/n = ln(a)/ln(b) = 1.3333333333333333
a/b = n/(n+1) = ln(b)/ln(a) = 0.75
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.083333333333333
b/a-a/b= (2n+1)/(n2+n) = 0.5833333333333333


Se n = 4 (Quatro):

n = 4
a = [n/(n+1)]n+1 = 0.3276800000000001
b = [n/(n+1)]n = 0.4096000000000001
aa= bb = 0.6937813717015524
b/a = (n+1)/n = ln(a)/ln(b) = 1.25
a/b = n/(n+1) = ln(b)/ln(a) = 0.8
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.05
b/a-a/b= (2n+1)/(n2+n) = 0.44999999999999996


Se n = 5 (Cinco):

n = 5
a = [n/(n+1)]n+1 = 0.3348979766803842
b = [n/(n+1)]n = 0.401877572016461
aa= bb = 0.693256843543474
b/a = (n+1)/n = ln(a)/ln(b) = 1.2
a/b = n/(n+1) = ln(b)/ln(a) = 0.8333333333333334
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.033333333333333
b/a-a/b= (2n+1)/(n2+n) = 0.3666666666666666


Se n = 6 (Seis):

n = 6
a = [n/(n+1)]n+1 = 0.3399166770891136
b = [n/(n+1)]n = 0.3965694566039659
aa= bb = 0.6929560456422869
b/a = (n+1)/n = ln(a)/ln(b) = 1.1666666666666667
a/b = n/(n+1) = ln(b)/ln(a) = 0.8571428571428571
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.0238095238095237
b/a-a/b= (2n+1)/(n2+n) = 0.30952380952380965

Para 1/n com n de 1 até 6


Se n = 1 :

n = 1
a = [n/(n+1)]n+1 = 0.25
b = [n/(n+1)]n = 0.5
aa= bb = 0.7071067811865476
b/a = (n+1)/n = ln(a)/ln(b) = 2
a/b = n/(n+1) = ln(b)/ln(a) = 0.5
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.5
b/a-a/b= (2n+1)/(n2+n) = 1.5


Se n = 0.5 :

n = 0.5
a = [n/(n+1)]n+1 = 0.19245008972987523
b = [n/(n+1)]n = 0.5773502691896257
aa= bb = 0.7282273028722097
b/a = (n+1)/n = ln(a)/ln(b) = 3
a/b = n/(n+1) = ln(b)/ln(a) = 0.3333333333333333
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.3333333333333335
b/a-a/b= (2n+1)/(n2+n) = 2.6666666666666665


Se n = 0.3333333333333333 :

n = 0.3333333333333333
a = [n/(n+1)]n+1 = 0.15749013123685918
b = [n/(n+1)]n = 0.6299605249474366
aa= bb = 0.7474382584741919
b/a = (n+1)/n = ln(a)/ln(b) = 4
a/b = n/(n+1) = ln(b)/ln(a) = 0.25
b/a+a/b= (2n2+2n+1)/(n2+n) = 4.25
b/a-a/b= (2n+1)/(n2+n) = 3.75


Se n = 0.25 :

n = 0.25
a = [n/(n+1)]n+1 = 0.1337480609952844
b = [n/(n+1)]n = 0.668740304976422
aa= bb = 0.7640867110627018
b/a = (n+1)/n = ln(a)/ln(b) = 5
a/b = n/(n+1) = ln(b)/ln(a) = 0.2
b/a+a/b= (2n2+2n+1)/(n2+n) = 5.2
b/a-a/b= (2n+1)/(n2+n) = 4.8


Se n = 0.2 :

n = 0.2
a = [n/(n+1)]n+1 = 0.1164711864619299
b = [n/(n+1)]n = 0.6988271187715793
aa= bb = 0.778469067871172
b/a = (n+1)/n = ln(a)/ln(b) = 5.999999999999999
a/b = n/(n+1) = ln(b)/ln(a) = 0.16666666666666669
b/a+a/b= (2n2+2n+1)/(n2+n) = 6.166666666666666
b/a-a/b= (2n+1)/(n2+n) = 5.833333333333332


Se n = 0.16666666666666666 :

n = 0.16666666666666666
a = [n/(n+1)]n+1 = 0.10328857519992624
b = [n/(n+1)]n = 0.7230200263994837
aa= bb = 0.7909751972584026
b/a = (n+1)/n = ln(a)/ln(b) = 7.000000000000001
a/b = n/(n+1) = ln(b)/ln(a) = 0.14285714285714285
b/a+a/b= (2n2+2n+1)/(n2+n) = 7.142857142857144
b/a-a/b= (2n+1)/(n2+n) = 6.857142857142858

Usando valores com Pi:

Se n = 3.141592653589793 pi:

n = 3.141592653589793
a = [n/(n+1)]n+1 = 0.3183731004122771
b = [n/(n+1)]n = 0.4197144057684896
aa= bb = 0.694621641492507
b/a = (n+1)/n = ln(a)/ln(b) = 1.3183098861837907
a/b = n/(n+1) = ln(b)/ln(a) = 0.7585469929947761
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.0768568791785667
b/a-a/b= (2n+1)/(n2+n) = 0.5597628931890146


Se n = 2.141592653589793 pi-1:

n = 2.141592653589793
a = [n/(n+1)]n+1 = 0.30005310705153293
b = [n/(n+1)]n = 0.4401605670526569
aa= bb = 0.6968377567491657
b/a = (n+1)/n = ln(a)/ln(b) = 1.46694220692426
a/b = n/(n+1) = ln(b)/ln(a) = 0.6816901138162094
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.1486323207404694
b/a-a/b= (2n+1)/(n2+n) = 0.7852520931080506


Se n = 0.3183098861837907 1/pi:

n = 0.3183098861837907
a = [n/(n+1)]n+1 = 0.15359678995017048
b = [n/(n+1)]n = 0.6361353368726007
aa= bb = 0.749947580554872
b/a = (n+1)/n = ln(a)/ln(b) = 4.141592653589793
a/b = n/(n+1) = ln(b)/ln(a) = 0.24145300700522385
b/a+a/b= (2n2+2n+1)/(n2+n) = 4.383045660595017
b/a-a/b= (2n+1)/(n2+n) = 3.9001396465845692


Se n = 0.46694220692425986 1/(pi-1):

n = 0.46694220692425986
a = [n/(n+1)]n+1 = 0.18651332679971905
b = [n/(n+1)]n = 0.5859488972705897
aa= bb = 0.7311014940062435
b/a = (n+1)/n = ln(a)/ln(b) = 3.1415926535897936
a/b = n/(n+1) = ln(b)/ln(a) = 0.31830988618379064
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.459902539773584
b/a-a/b= (2n+1)/(n2+n) = 2.823282767406003

Usando valores com e:


Se n = 2.718281828459045 e:

n = 2.718281828459045
a = [n/(n+1)]n+1 = 0.31198653777786745
b = [n/(n+1)]n = 0.4267599709486024
aa= bb = 0.6953079526442089
b/a = (n+1)/n = ln(a)/ln(b) = 1.3678794411714423
a/b = n/(n+1) = ln(b)/ln(a) = 0.7310585786300049
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.098938019801447
b/a-a/b= (2n+1)/(n2+n) = 0.6368208625414374


Se n = 1.718281828459045 e-1:

n = 1.718281828459045
a = [n/(n+1)]n+1 = 0.28742069137944415
b = [n/(n+1)]n = 0.454692838834558
aa= bb = 0.6988224971233713
b/a = (n+1)/n = ln(a)/ln(b) = 1.5819767068693265
a/b = n/(n+1) = ln(b)/ln(a) = 0.6321205588285577
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.214097265697884
b/a-a/b= (2n+1)/(n2+n) = 0.9498561480407688


Se n = 0.36787944117144233 1/e:

n = 0.36787944117144233
a = [n/(n+1)]n+1 = 0.16589774685298547
b = [n/(n+1)]n = 0.6168545775057546
aa= bb = 0.7422894613778632
b/a = (n+1)/n = ln(a)/ln(b) = 3.718281828459045
a/b = n/(n+1) = ln(b)/ln(a) = 0.2689414213699951
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.9872232498290403
b/a-a/b= (2n+1)/(n2+n) = 3.44934040708905


Se n = 0.5819767068693265 1/(e-1):

n = 0.5819767068693265
a = [n/(n+1)]n+1 = 0.2055683479587981
b = [n/(n+1)]n = 0.5587927047627469
aa= bb = 0.7223797288294247
b/a = (n+1)/n = ln(a)/ln(b) = 2.718281828459045
a/b = n/(n+1) = ln(b)/ln(a) = 0.36787944117144233
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.0861612696304874
b/a-a/b= (2n+1)/(n2+n) = 2.3504023872876028

Usando valores com o número aureo (x tal que x2=x+1):


Se n = 1.618033988749895 ouro:

n = 1.618033988749895
a = [n/(n+1)]n+1 = 0.2837025599424474
b = [n/(n+1)]n = 0.4590403846822343
aa= bb = 0.699480971720873
b/a = (n+1)/n = ln(a)/ln(b) = 1.618033988749895
a/b = n/(n+1) = ln(b)/ln(a) = 0.6180339887498949
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.23606797749979
b/a-a/b= (2n+1)/(n2+n) = 1


Se n = 0.6180339887498949 ouro-1:

n = 0.6180339887498949
a = [n/(n+1)]n+1 = 0.21071807476921361
b = [n/(n+1)]n = 0.5516670817897429
aa= bb = 0.7202640573833972
b/a = (n+1)/n = ln(a)/ln(b) = 2.618033988749895
a/b = n/(n+1) = ln(b)/ln(a) = 0.38196601125010515
b/a+a/b= (2n2+2n+1)/(n2+n) = 3
b/a-a/b= (2n+1)/(n2+n) = 2.23606797749979


Se n = 0.6180339887498948 1/ouro:

n = 0.6180339887498948
a = [n/(n+1)]n+1 = 0.21071807476921356
b = [n/(n+1)]n = 0.5516670817897429
aa= bb = 0.7202640573833972
b/a = (n+1)/n = ln(a)/ln(b) = 2.6180339887498953
a/b = n/(n+1) = ln(b)/ln(a) = 0.3819660112501051
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.0000000000000004
b/a-a/b= (2n+1)/(n2+n) = 2.2360679774997902


Se n = 1.6180339887498947 1/(ouro-1):

n = 1.6180339887498947
a = [n/(n+1)]n+1 = 0.2837025599424473
b = [n/(n+1)]n = 0.45904038468223424
aa= bb = 0.699480971720873
b/a = (n+1)/n = ln(a)/ln(b) = 1.6180339887498951
a/b = n/(n+1) = ln(b)/ln(a) = 0.6180339887498948
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.23606797749979
b/a-a/b= (2n+1)/(n2+n) = 1.0000000000000004

Usando valores com x tal que x3=x+1:


Se n = 1.3247179572447463 x:

n = 1.3247179572447463
a = [n/(n+1)]n+1 = 0.2705172532293568
b = [n/(n+1)]n = 0.4747246860265993
aa= bb = 0.7020999508903193
b/a = (n+1)/n = ln(a)/ln(b) = 1.7548776662466927
a/b = n/(n+1) = ln(b)/ln(a) = 0.5698402909980533
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.3247179572447463
b/a-a/b= (2n+1)/(n2+n) = 1.1850373752486394


Se n = 0.3247179572447463 x-1:

n = 0.3247179572447463
a = [n/(n+1)]n+1 = 0.15527607328946827
b = [n/(n+1)]n = 0.6334635890246503
aa= bb = 0.7488552679286654
b/a = (n+1)/n = ln(a)/ln(b) = 4.079595623491437
a/b = n/(n+1) = ln(b)/ln(a) = 0.2451223337533074
b/a+a/b= (2n2+2n+1)/(n2+n) = 4.324717957244744
b/a-a/b= (2n+1)/(n2+n) = 3.834473289738129


Se n = 0.7548776662466926 1/x:

n = 0.7548776662466926
a = [n/(n+1)]n+1 = 0.22754379395582508
b = [n/(n+1)]n = 0.528975143868705
aa= bb = 0.7140097294066174
b/a = (n+1)/n = ln(a)/ln(b) = 2.3247179572447463
a/b = n/(n+1) = ln(b)/ln(a) = 0.4301597090019467
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.754877666246693
b/a-a/b= (2n+1)/(n2+n) = 1.8945582482427996


Se n = 3.0795956234914366 1/(x-1):

n = 3.0795956234914366
a = [n/(n+1)]n+1 = 0.31753076975351735
b = [n/(n+1)]n = 0.42063871267023134
aa= bb = 0.6947069868601304
b/a = (n+1)/n = ln(a)/ln(b) = 1.3247179572447463
a/b = n/(n+1) = ln(b)/ln(a) = 0.7548776662466926
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.079595623491439
b/a-a/b= (2n+1)/(n2+n) = 0.5698402909980537

Usando valores com x tal que x4=x+1:


Se n = 1.2207440846057596 x:

n = 1.2207440846057596
a = [n/(n+1)]n+1 = 0.264780870793223
b = [n/(n+1)]n = 0.48168208222013253
aa= bb = 0.7033822636267789
b/a = (n+1)/n = ln(a)/ln(b) = 1.8191725133961645
a/b = n/(n+1) = ln(b)/ln(a) = 0.5497004779019703
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.368872991298135
b/a-a/b= (2n+1)/(n2+n) = 1.2694720354941942


Se n = 0.22074408460575956 x-1:

n = 0.22074408460575956
a = [n/(n+1)]n+1 = 0.12396810595944582
b = [n/(n+1)]n = 0.6855600787674515
aa= bb = 0.7719680989413776
b/a = (n+1)/n = ln(a)/ln(b) = 5.530132718101876
a/b = n/(n+1) = ln(b)/ln(a) = 0.18082748660383563
b/a+a/b= (2n2+2n+1)/(n2+n) = 5.710960204705711
b/a-a/b= (2n+1)/(n2+n) = 5.3493052314980405


Se n = 0.8191725133961644 1/x:

n = 0.8191725133961644
a = [n/(n+1)]n+1 = 0.2342391198275361
b = [n/(n+1)]n = 0.5201851397402604
aa= bb = 0.7117862252181024
b/a = (n+1)/n = ln(a)/ln(b) = 2.2207440846057596
a/b = n/(n+1) = ln(b)/ln(a) = 0.45029952209802976
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.6710436067037895
b/a-a/b= (2n+1)/(n2+n) = 1.7704445625077299


Se n = 4.530132718101876 1/(x-1):

n = 4.530132718101876
a = [n/(n+1)]n+1 = 0.3318595407174712
b = [n/(n+1)]n = 0.4051155712508372
aa= bb = 0.6934643133921594
b/a = (n+1)/n = ln(a)/ln(b) = 1.2207440846057596
a/b = n/(n+1) = ln(b)/ln(a) = 0.8191725133961644
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.039916598001924
b/a-a/b= (2n+1)/(n2+n) = 0.4015715712095952

Usando valores com k tal que kk=e:


Se n = 1.7632228343518932 x:

n = 1.7632228343518932
a = [n/(n+1)]n+1 = 0.2889822471758976
b = [n/(n+1)]n = 0.45287658970924993
aa= bb = 0.6985561772585763
b/a = (n+1)/n = ln(a)/ln(b) = 1.567143290409785
a/b = n/(n+1) = ln(b)/ln(a) = 0.6381037433651103
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.2052470337748953
b/a-a/b= (2n+1)/(n2+n) = 0.9290395470446747


Se n = 0.7632228343518932 x-1:

n = 0.7632228343518932
a = [n/(n+1)]n+1 = 0.22845133911325144
b = [n/(n+1)]n = 0.5277758991642433
aa= bb = 0.7136997817812418
b/a = (n+1)/n = ln(a)/ln(b) = 2.310233335522739
a/b = n/(n+1) = ln(b)/ln(a) = 0.432856709590215
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.7430900451129543
b/a-a/b= (2n+1)/(n2+n) = 1.8773766259325242


Se n = 0.567143290409785 1/x:

n = 0.567143290409785
a = [n/(n+1)]n+1 = 0.20334711713751544
b = [n/(n+1)]n = 0.5618933973740118
aa= bb = 0.7233228679199418
b/a = (n+1)/n = ln(a)/ln(b) = 2.763222834351893
a/b = n/(n+1) = ln(b)/ln(a) = 0.36189625663488967
b/a+a/b= (2n2+2n+1)/(n2+n) = 3.1251190909867828
b/a-a/b= (2n+1)/(n2+n) = 2.4013265777170036


Se n = 1.3102333355227391 1/(x-1):

n = 1.3102333355227391
a = [n/(n+1)]n+1 = 0.269757105125455
b = [n/(n+1)]n = 0.4756418874858662
aa= bb = 0.7022647905016444
b/a = (n+1)/n = ln(a)/ln(b) = 1.7632228343518932
a/b = n/(n+1) = ln(b)/ln(a) = 0.5671432904097851
b/a+a/b= (2n2+2n+1)/(n2+n) = 2.330366124761678
b/a-a/b= (2n+1)/(n2+n) = 1.1960795439421081

Alguns b/a+a/b:

Se n=1/1 = (22+1) / 2 = 2.5
Se n=1/2 = (32+1) / 3 = 3.3333333333333335
Se n=1/3 = (42+1) / 4 = 4.25
Se n=1/4 = (52+1) / 5 = 5.2
Se n=1/5 = (62+1) / 6 = 6.166666666666667
Se n=1/6 = (72+1) / 7 = 7.142857142857143
Se n=1/7 = (82+1) / 8 = 8.125
Se n=1/8 = (92+1) / 9 = 9.11111111111111
Se n=1/9 = (102+1) / 10 = 10.1
Se n=1.618033988749895 = b/a+a/b = 2.23606797749979
Se n=1.3247179572447463 = b/a+a/b = 2.3247179572447454
Se n=1.2207440846057596 = b/a+a/b = 2.368872991298135
Se n=1/1.618033988749895 = b/a+a/b = 3
Se n=1/1.3247179572447463 = b/a+a/b = 2.754877666246693
Se n=1/1.2207440846057596 = b/a+a/b = 2.6710436067037895

Alguns b/a-a/b:

Se n=1/1 = (22-1) / 2 = 1.5
Se n=1/2 = (32-1) / 3 = 2.6666666666666665
Se n=1/3 = (42-1) / 4 = 3.75
Se n=1/4 = (52-1) / 5 = 4.8
Se n=1/5 = (62-1) / 6 = 5.833333333333333
Se n=1/6 = (72-1) / 7 = 6.857142857142857
Se n=1/7 = (82-1) / 8 = 7.875
Se n=1/8 = (92-1) / 9 = 8.88888888888889
Se n=1/9 = (102-1) / 10 = 9.9
Se n=1.618033988749895 = b/a-a/b = 1
Se n=1.3247179572447463 = b/a-a/b = 1.1850373752486392
Se n=1.2207440846057596 = b/a-a/b = 1.269472035494194
Se n=1/1.618033988749895 = b/a-a/b = 2.23606797749979
Se n=1/1.3247179572447463 = b/a-a/b = 1.8945582482427998
Se n=1/1.2207440846057596 = b/a-a/b = 1.77044456250773


























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