Lösungen / Solutions

1

a

Achtung Summationbereich!/  Attention aux limits de l'index!

Sum[1/Pi^n, {n, 0, Infinity}]

1/(1 - 1/Pi) // Simplify

Sum[1/Pi^n, {n, 1, Infinity}]

(1 + Sum[1/Pi^n, {n, 1, Infinity}])

(1 + Sum[1/Pi^n, {n, 1, Infinity}]) (Pi - 1) // 
      Together

b

Erklärungsbeispiele: / Exemples pour l'explication:

Sum[1/k^2 - 1/(k + 1)^2, {k, a, 100}]

Bildung der Summe: / Construction de la somme:

Sum[1/(a + k)^2 - 1/(a + k + 1)^2, {k, 1, 100}]

Sum[1/(a + k)^2 - 1/(a + k + 1)^2, {k, 1, Infinity}]

1/a^2 - 1/(a + 1)^2 + 1/(a + 1)^2 - 1/(a + 2)^2 + 
      1/(a + 2)^2 - 1/(a + 3)^2 + 1/(a + 3)^2 - 
      1/(a + 4)^2

Sum[1/(a + k)^2 - 1/(a + k + 1)^2, {k, 1, Infinity}] /. 
      a -> 0

1/1^2 - 1/2^2 + 1/2^2 - 1/3^2 + 1/3^2 - 1/4^2 + 1/4^2 - 1/5^2 + 1/5^2 - 
        1/6^2 + 1/6^2 - 1/7^2 + 1/7^2 - 1/8^2 + 1/8^2 - 1/9^2 + 1/9^2 - 
        1/10^2
 // N

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

2

a

f[x_] := 5  x^3 - 3  x^2 - 2  x - 4 + Sqrt[x];      f[
      x]

5  x^3 - 3  x^2 - 2 - 4 + x^(1/2)

D[f[x], x]

D[f[x], x] /. x -> 1

% // N

D[f[x], x] /. x -> 0

b

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

D[f[x], x]

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

c

D[1/Sqrt[x], x]

f[x_] := 1/(Sqrt[x] - x);    f[x]

D[f[x], x]

D[f[x], x]  //  Together

d

D[Cos[x^2 + 1], x]

D[Cos[x]^2 + 1, x]

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

D[f[x], x]

3

f[x_] := x^2 - x + 1;    f[x]

D[f[x], x]

ArcTan[D[f[x], x]]

ArcTan[D[f[x], x]] /. x -> 1

360 %/(2  Pi)