Lösungen - Solutions

1

Remove["Global`*"]

Solve[(24 - y)/y == 40/24, {y}]

xx[0] = 24;

xx[1] = xx[0] - y /.  y -> 9

Solve[y/xx[1]  == xx[1] /24, {y}]

xx[2] = y /. y -> (75/8)

xx[1] ^2/24

xx[1]/xx[0]

xx[2]/xx[1]

xx[n_] := xx[n - 1]  xx[1]/xx[0];
Table[xx[k], {k, 0, 10}]

xx[n_] := 24  ((5/8))^n;
Table[xx[k], {k, 0, 10}]

aQ = 2*24*(1/(1 - 5/8))

aQ = 2*Sum[24 ((5/8))^n, {n, 0, Infinity}]

2

Remove["Global`*"]

f[x_] := (x^4 + 1) (x^2 - 1) + 1

a

D[f[x], x]

D[f[x], x] // Expand

b

% /. x -> 1

c

Solve[(D[f[x], x] /. x -> 1) == 
        Tan[α], {α}]

% // N

3

Remove["Global`*"]

f[x_] := (4 x^5 - 6  x^4 + 2  x^3 - x^2) 2/x^2

a

D[f[x], {x, 2}] // Expand

Solve[% == 10, {x}]

b

Solve[%% == 0, {x}]

4

Remove["Global`*"]

f[x_] := (a x  +  b) (x + c)

Solve[{f[0] == 1, f[1] == -(a c  +  b),  
        Evaluate[D[f[x], x] /. x -> 1] == 0}, {a, b, 
        c}]

Keine reelle Lösung

5

Remove["Global`*"]

a

a[n_] := 
        Sum[(k + 1)/(2 k^3  + 1), {k, 1, 
            n}];
Table[a[n], {n, 1, 50}] // N

Konvergent: Nenner kubisch, Zähler nur linear, Graddifferenz 2

b

Sum[1/(n^2 - 1), {n, 2, Infinity}]

Teleskopreihe

6

Remove["Global`*"]

a

c[
          n_] := (4 Sin[5  n - n^2] + 2 n^5  -  6 n)/(3  +  
              a n^7 -  b n^5);
c[n]

1) a unequal 0 ===> c[n] conv. to 0
2) a = 0 AND b unequal 0 ===> c[n] conv. to  - 2 / b
3) a = 0 AND b = 0 ===> c[n] conv. to infinity

===> Limit[c[n], n -> Infinity]= 4  ===> Case 2):

a = 0, - 2 / b= 4,  b = -1/2

7

Remove["Global`*"]

a[1] = 3;
a[2] = -5;
b[n_] := a[n - 1];
a[n_] := a[n - 1] + b[n - 1];
{a[7], b[7]}