Lösungen - Solutions

1

Remove["Global`*"]

a[0] = 10; b[0] = 20; c[0] = Sqrt[10^2 + 20^2]

c[1] = a[0];
Solve[c[0]/a[0] == c[1]/a[1], {a[1]}]

a[1] = a[1] /. a[1] -> 2 (Sqrt[5])

Solve[a[0]/a[1] == b[0]/b[1], {b[1]}]

b[1] = b[1] /. b[1] -> 4 (Sqrt[5])

ch[2] = b[1];
Solve[a[1]/c[1] == bh[2]/ch[2], {bh[2]}]

c[2] = bh[2] /. bh[2]->4

fak = c[2]/c[1]

aTotal = 20*10 /2

aWeiss[0] = b[1] c[1]/2

aWeiss[1] = aWeiss[1] fak

aWeiss[n_] := aWeiss[0] fak^n

aWeissAll =  Sum[aWeiss[n], {n, 0, Infinity}]

aWeissAll = aWeiss[0] 1/(1 - fak)

aSchwarzAll = aTotal - aWeissAll

aSchwarzAll // N

2

Remove["Global`*"]

f[x_] := (x^2 + 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_] := 2 x^3 - 3  x^2 + 4  x - 6

a

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

Solve[% == 10, {x}]

b

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

4

Remove["Global`*"]

f[x_] := (a x^2  +  b x  + c)

Solve[{f[-2] == 0, 
        Evaluate[D[f[x], x] /. x -> 3] == 2}, {a, b, 
        c}]

Keine reelle Lösung, eine Gleichung fehlt

Solve[{f[-2] == 0, f[0] == 1, 
        Evaluate[D[f[x], x] /. x -> 3] == 2}, {a, b, 
        c}]

5

Remove["Global`*"]

a

a[n_] := 
        Sum[(2  k + 3)/(3 k^3  + 2), {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_] := (2 Cos[n^2 - 1] + 4 n^3  -  5 n)/(a n^4 +  
              b n^2);
c[n]

1) a unequal 0 ===> c[n] conv. to 0
2) a = 0 AND b unequal 0 ===> c[n] div.
3) a = 0 AND b = 0 ===> c[n] not def.

===> Limit[c[n], n -> Infinity]= 4  not possible

7

Remove["Global`*"]

a[1] = 3;
a[2] = -5;
b[n_] := a[n - 1];
a[n_] := a[n - 1] + 2  b[n - 1];
{a[10], b[10]}