Lösungen/Solutions/E1S1AnUe/Ex4

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

a

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

b

Interessant, wie Mathematica reagiert:

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

Limit[(-1)^n  Product[k/n, {k, 1, n}] , 
      n -> Infinity]

Limit[(-1)^n  Product[k/n, {k, 2, n}] 1/n, 
      n -> Infinity]

Limit[1/n, n -> Infinity]

Limit[(-1)^n  Product[k/n, {k, 2, n}] , 
      n -> Infinity]

(1/n)   (2/n) (3/n) 
                (4/n)  ...  ...  ((n - 1)/n)  =  p  <  1

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

Limit[1/n, n -> Infinity]

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

Ueb/Ex 2      W

a

FactorInteger[6]

FactorInteger[20]

FactorInteger[42]

FactorInteger[72]

FactorInteger[6]

FactorInteger[20 - 2]

FactorInteger[42 - 2^2 3]

20 - 6

42 - 20

22 - 14

72 - 42

30 - 22

2. Differenzenfolge konstant-> 1. linear ->0. quadratisch

a[n_] := a n^2  +  b n  +  c

solv = 
      Solve[{a[1] == 6, a[2] == 20, a[3] == 42}, {a, b, 
            c}] // Flatten

a1[n_] := a[n] /.  solv ;  a1[n]

Table[1/a1[n], {n, 1, 4}]

b

Table[1 + Sum[2  k - 1, {k, 1, n}], {n, 1, 5}]

Sum[2  k - 1, {k, 1, n}] // Expand

b[n_] := n^2 + 1

Table[1/Sqrt[b[n]], {n, 1, 4}]

Ueb/Ex 3      W

a

(Sqrt[n + 1] - Sqrt[n]) (Sqrt[n + 1] + Sqrt[n]) // 
      Expand

(Sqrt[n + 1] - Sqrt[n])  == 
      1/(Sqrt[n + 1] + Sqrt[n])

1/(Sqrt[n + 1] + Sqrt[n]) >=   1/(Sqrt[n + 2] + 
            Sqrt[n + 1])

Rechts Nenner grösser, streng monoton fallend.
Beschränkt, da Ausdruck zwischen 0 und 1

b

(n^2 + 1)/n  // Apart

Mit n streng monoton wachsend, nicht beschränkt.

Ueb/Ex 4      W

a

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

b

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

c

Limit[Sum[5 (1/3)^(k - 1), {k, 0, n}], 
      n -> Infinity]

d

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

Ueb/Ex 5      W

a

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

b

Remove[x]

solve4 = 
      Solve[99 x  ==  
            3.0404040404040404   -  0.0404040404040404  , {x}] // 
        Flatten

x = x /. solve4  // Rationalize

% // N