Die Zahl e, Grenzwerte, Kettenbrüche / Le nombre e, valeurs limite, fractions continues

Entwicklung von e / Développer e

f[n_]:=(1 + 1/n)^n;Table[{n, f[n]}, {n, 1, 100}] // N  // MatrixForm

E // N

N[E, 100]

f[n_] := (1 + 2/n)^(n/2 *2);Table[{n, f[n]}, {n, 1, 100}] // N  // MatrixForm

f[n_] := ((1 + 2/n)^(n/2 ))^2;Table[{n, f[n]}, {n, 1, 100}] // N  // MatrixForm

E^2  // N

f[n_] := (1 + 2/n)^(n/2); Table[{n, f[n]}, {n, 1, 100}] // 
        N  // MatrixForm

E // N

f[n_] := (1 - 1/n)^n; Table[{n, f[n]}, {n, 1, 100}] // N  // 
      MatrixForm

Sum[1/n!, {n, 0, Infinity}]

Sum[1/n!, {n, 0, 100}] // N

(N[Sum[1/n!, {n, 0, 100}]] - E)  // N

Grenzwerte / Valeurs limite

Limit[(1 + 1/n)^n, n -> Infinity]

Limit[(1 + 1/n)^n, n -> Infinity] // N

Limit[(Cos[n]^2 + n^3 - (n - 2)^5)/n^1, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + (n - 2)^5)/n^1, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + (n - 2)^5)/n^6, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + (n - 2)^5)/n^5, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + (4  n - 2)^5)/n^5, 
      n -> Infinity]

4^5

Limit[(Cos[n]^2 + n^3 + 4 (n - 2)^5)/n^5, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + k (n - 2)^5)/n^5, 
      n -> Infinity]

Limit[(Cos[n]^2 + n^3 + k (n - 2)^5)/n^5, n -> Infinity] /. 
      k -> 2

Limit[(Cos[n]^2 + n^3 + (k n - 2)^5)/n^5, n -> Infinity] /. 
      k -> 2

2^5

Limit[Sin[1/n]/(1/n), n -> Infinity]

Limit[Sin[2/n]/(1/n), n -> Infinity]

Limit[Sin[1/n]/(3/n), n -> Infinity]

Kettenbrüche / Fractions continues

Umgang mit der der Kettenbruchentwicklung / Traitements de fractions continues

?ContinuedFraction

ContinuedFraction[(35/16)]

ContinuedFraction[(1/3) (1 + (Sqrt[2]))]

ContinuedFraction[(E^2), 27]

ContinuedFraction[E, 27]

ContinuedFraction[E, 100]

?Continu*

?ContinuedFraction

ContinuedFraction[E]

?FromContinuedFraction

a + (1/(b + 1/(c + 1/e)))

a + (b/(c + d)) // Together

a + (b/(c + d/(e + f))) // Together

a + (b/(c + d/(e + f/(g + h)))) // Together

Solve[A == (a + 1/(x)), x]

Solve[x == Floor[x] + r, r]

Solve[r == 1/x, x]

Programmierung der Kettenbruchentwicklung

Remove[x]

x[n_] := 
        1/(x[n - 1] - Floor[x[n - 1]]);
tab[z_, k_] := {x[0] = z, 
          Prepend[Table[ Floor[x[n]], {n, 1, k}], 
            Floor[x[0]]]};
tab1[z_, k_] := 
        Delete[{x[0] = z, 
              Prepend[Table[ Floor[x[n]], {n, 1, k}], Floor[x[0]]]}, 1] // 
          Flatten;

tab[E, 10]

tab[E, 12]

ContinuedFraction[E, 13]

tab1[E, 12]

FromContinuedFraction[%]

u1 = N[FromContinuedFraction[tab1[E, 12]], 30]

ContinuedFraction[E, 100]

u2 = N[FromContinuedFraction[%], 30]

u1 - u2

More Work

Limit[(1+1/n)^n,n->Infinity]

?*Fraction*

ContinuedFraction[E]

ContinuedFraction[E,200]

ContinuedFraction[4/7]

ContinuedFraction[Pi,400]

N[Pi,1000]

ContinuedFraction[4/7]

?Sum

Sum[1/k,{k,1,Infinity}]

Limit[Sum[1/k,{k,1,n}],n->Infinity]

Limit[Sum[1/k^2,{k,1,n}],n->Infinity]

Sum[1/k^2,{k,1,Infinity}]

Sum[1/k^3,{k,1,n}]

Sqrt[2]!

5!

Sum[1/k^3,{k,1,5}]

Sum[1/k^3,{k,1,Infinity}]

Remove["Global`*"]

Limit[Sum[1/k^3,{k,1,n}],n->Infinity]