Lösungen / Solutions

1

a

s[n_] := Sum[1/2^k - 1/3^k, {k, 1, n}]
Limit[s[n], n -> Infinity]

b

s[n_] := Sum[1/3^k - 1/4^k, {k, 1, n}]
Limit[s[n], n -> Infinity]

2

a

Table[Sum[1/3^k, {k, 1, n}], {n, 1, 10}]

Solve[Sum[1/3^k, {k, 1, n}] == (3280/6561), {n}]

b

Table[Sum[1/4^k, {k, 1, n}], {n, 1, 10}]

Solve[Sum[1/4^k, {k, 1, n}] == (5461/16384), {n}]

3

a

Limit[(2  Abs[Sin[1/n]] n^3  + 2  n^3 - n + 1)/(4  n^3 - 
            n^2), {n -> Infinity}]

b

Limit[(3  Abs[Sin[1/n]] n^3  - 3  n^3 - 2  n - 1)/(2  n^3 - 
            n^2), {n -> Infinity}]

4

a

Limit[(Abs[Tan[n]]/n  + 5  n^2 - 3  n)/(2  n^3 - 
            3), {n -> Infinity}]

Limit[
      Sin[n]/(n Cos[n] (2  n^2 - 3)) , {n -> 
          Infinity}]

Limit[
      Abs[1/Sqrt[n^2/Sin[n]^2 - n^2]]/(2  n^2 - 3)  + (5  n^2 - 
              3  n)/(2  n^3 - 3), {n -> Infinity}]

b

Limit[(Abs[Tan[n]]/n  - 5  n^2 - 3  n)/(4  n^3 - 
            3), {n -> Infinity}]

Limit[Abs[Sin[n]/Sqrt[1 - Sin[n]^2]]/
            n/(4  n^2 - 3)  + (5  n^2 - 3  n)/(4  n^2 - 
              3), {n -> Infinity}]

Limit[Abs[1/Sqrt[1 - Sin[n]^2]]/
            n/(4  n^2 - 3)  + (5  n^2 - 3  n)/(4  n^2 - 
              3), {n -> Infinity}]

5

a

Limit[(E^n)/(Log[n] + E^(3  n)) + 
        n^4/E^n, {n -> Infinity}]

b

Limit[(E^n)/(Log[n] - E^(2  n)) - 
        n^3/E^n, {n -> Infinity}]

6

a[1] = 1;  a[2] = 1;  a[n_] := a[n - 1] + a[n - 2];  
    q[n_] := a[n + 1]/a[n]

Table[a[n], {n, 1, 10}]

(b[n] + b[n - 1])/b[n]  //  Apart

0  <  b[-1 + n]/b[n]  <  1

a

N[Table[q[2  n], {n, 1, 8}], 10]

b

N[Table[q[2  n + 1], {n, 1, 8}], 10]

7 Monoton - stetig - beschränkt

f[x_] := Floor[Sin[x]];
g[x_] := Floor[Cos[x]];

Plot[f[x], {x, 0, 4  Pi}]

Plot[f[x], {x, -Pi, Pi}];

Plot[f[x], {x, -10, 10}];

Plot[g[x], {x, 0, 4  Pi}]

Plot[g[x], {x, -10, 10}];

8

f[x_] := x Sin[1/x];
g[x_] := x Cos[1/x];

Plot[f[x], {x, -Pi, Pi}];

Plot[f[x], {x, -10, 10}];

Plot[g[x], {x, 0, 4  Pi}];

Plot[g[x], {x, -10, 10}];

9

f[x_] := Log[1 + Sin[x]^2];
g[x_] := Log[1 + Cos[x]^2];

Plot[f[x], {x, -5 Pi, 5  Pi}];

Plot[g[x], {x, -5 Pi, 5  Pi}];