Lösungen / Solutions/I1S2Ue/Ex15

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

Ueb/Ex 1      W

a    (Subscript[a, n])(x) = ?

?Series

Series[Log[x],  {x,  1,  20}]

a[x_, 
          n_] := (((-1))^((n + 1)))*(1/n) *(((x - 1))^n);
a[x, 10]

b Konvergenzradius / Rayon de convergence

Quotientenkriterium / Critère de d'Alembert

a[x, n + 1]/a[x, n]

r00[x_] := Limit[a[x, n + 1]/a[x, n], n -> Infinity]; 
    r00[x]

===>    |1-x| = q < 1       für Konv. / pour la conv.

Plot[{Abs[1 - x], 1}, {x, -1, 3}];

I=[0,2],         === >      R=1

Ueb/Ex 2      W

a    (Subscript[a, n])(x) = ?

Series[E^(-x^2),  {x,  0,  20}]

a[x_, n_] := (-1)^(n + 1)*(1/(n!)) *
          (x^(2*n));
a[x, 10]

Series[E^(-x^2),  {x,  0,  20}] // Normal

b Konvergenzradius / Rayon de convergence

Quotientenkriterium / Critère de d'Alembert

a[x, n + 1]/a[x, n]

-a[x, n + 1]/
        a[x, n] /.  {(n!/(n + 1)!) -> 
          1/(n + 1)}

Simplify[-a[x, n + 1]/
          a[x, n]]  /.  {(n!/(n + 1)!) -> 
          1/(n + 1)}

r00[x_] := Limit[((x^2)/(1 + n)), n -> Infinity]; 
    r00[x]

===>    |r00[x]| = 0  = q < 1       für Konv. / pour la conv.
===>   Für alle Werte von  x  erfüllt!   / Satisfait pour tous les valeurs de x

=== >      R= ∞

c

Integrate[1-(x^2)+((x^4)/2)-((x^6)/6)+((x^8)/24)-((x^10)/120)+((x^12)/720)-((x^14)/5040)+((x^16)/40320)-((x^18)/362880)+((x^20)/3628800)+O[x]^21,x]

Integrate[a[x, n], x]

d

Integrate[1 - (x^2) + ((x^4)/2) - ((x^6)/6), x]

Integrate[1 - (x^2) + ((x^4)/2) - ((x^6)/6), {x, 0, 1}] // N

NIntegrate[E^(-x^2), {x, 0, 1}]

e

g[x_] := E^(-x^2)

Plot[g[x], {x, -3, 3}];

rR[n_, x_, h_] := 
      h^(n + 1)/(n + 1)!*  Evaluate[
          D[g[x], {x, n + 1}]]

rR[n, x, h]

rR[3, x, h]

rR3[x0_, μ_, h_] := 
        rR[3, x, h] /. 
          x -> (x0 + μ*h);
rR3[x0, μ, h]

rR3[x0, μ, 1]

rR3[0, μ, h]

Plot3D[rR3[0, μ, h], {μ, 0, 1}, {h, 0, 1}, 
        PlotRange -> {-0.5, 0.5}];

{rR3[0, 0, 0], rR3[0, 0, 1], rR3[0, 1, 0], rR3[0, 1, 1]} // 
      N

rR3[0, 0, 1]

Ueb/Ex 3      W