Potenzreihen Séries de puissances

Excursion: Séries de puissances etc. / Ausflug: Potenzreihen u.s.w.

Series[Tan[x],{x,0,20}]

f[x_]:=Normal[Series[Tan[x],{x,0,20}]];f[x]

Plot[Evaluate[{f[x],Tan[x]}],{x,-Pi/2+0.1,Pi/2 -0.1}];

Sum[Sin[n x],{n,0,100}]

Plot[Evaluate[{Sum[Sin[n x /Pi 10]/n,{n,1,1000}]}],{x,-2,2}];

Plot[Evaluate[{Sum[Cos[n x /Pi 10]/n -Sin[n x /Pi 10]/n,{n,1,1000}]}],{x,-2,2}];

Plot[Evaluate[{Sum[Cos[n x /Pi 10]/n,{n,1,1000}]}],{x,-2,2}];

Plot[Evaluate[{Sum[Cos[n x /Pi 10]/n +Sin[n x /Pi 10]/n,{n,1,1000}]}],{x,-2,2}];

Series[1/Tan[x],{x,0,20}]

Plot[Evaluate[{Normal[Series[1/Tan[x],{x,0,20}]],1/Tan[x]}],{x,-5,5}];

Plot[Evaluate[{Normal[Series[Log[x],{x,0,20}]],Log[x]}],{x,-5,5}];

Normal[Series[Log[x],{x,0,20}]]

Normal[Series[Log[x],{x,2,20}]]

Normal[Series[h[x],{x,x0,5}]]

f[x]

Normal[f[x]]/.x->1/5

a=N[(Normal[f[x]]/.x->1/5 ),30]

b=N[Tan[1/5],30]

a-b

Plot[E^-x^2,{x,-3,3}];

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

Plot[Evaluate[{E^-x^2,Normal[Series[E^-x^2,{x,0,20}]]}],{x,-2.2,2.2}];

Plot[Evaluate[{E^-x^2,Normal[Series[E^-x^2,{x,0,40}]]}],{x,-3,3}];

2

r=0.1

Clear[f];f[x]

f[x_]:=1/x;

α[x_]:=ArcTan[-1/D[f[x],x]];

dX[x_]:=r Cos[α[x]];
dY[x_]:=r Sin[α[x]];

m[x_]:={x,f[x]}+ {dX[x],dY[x]}

p1=ParametricPlot[{x,f[x]},{x,0.3,3},AspectRatio->Automatic];

Evaluate[(m[x]/.x->0.4)+{Cos[φ],Sin[φ]}]

p2=ParametricPlot[Evaluate[(m[x]/.x->0.4)+r{Cos[φ],Sin[φ]}],{φ,0,2Pi},AspectRatio->Automatic];

p3=ParametricPlot[Evaluate[(m[x]/.x->0.5)+r{Cos[φ],Sin[φ]}],{φ,0,2Pi},AspectRatio->Automatic];

p4=ParametricPlot[Evaluate[(m[x]/.x->0.6)+r{Cos[φ],Sin[φ]}],{φ,0,2Pi},AspectRatio->Automatic];

Show[p1,p2,p3,p4];

?Circle

Plot[Sin[x],{x,0,2π},Epilog->{PointSize[0.05],Hue[1],Point[{π,Sin[π]}]}];RGBColor[0,1,1]

newPlot[r_,z_,a_,b_]:=Plot[f[x],{x,a,b},Epilog->{{Hue[1],Disk[Evaluate[(m[x]/.x->z),r]]}},AspectRatio->Automatic];

newPlot[0.1,0.4,0.3,3];

Table[newPlot[0.1,s,0.3,3];,{s,0.3,3,0.2}]

?Mod

r=0.2;
f[x_]:=Sin[x];
α[x_]:=Mod[(ArcTan[-1/D[f[x],x]]+0.001)/Pi,1] Pi;
dX[x_]:=r Cos[α[x]];
dY[x_]:=r Sin[α[x]];
m[x_]:={x,f[x]}+ {dX[x],dY[x]};
newPlot[r_,z_,a_,b_]:=Plot[f[x],{x,a+z,b+z},Epilog->{{Hue[1],Disk[Evaluate[(m[x]/.x->z),r]]}},AspectRatio->Automatic,Axes->None,PlotRange->{-1.4,1.4}];

newPlot[0.1,0.4,0.3,3];

Table[newPlot[0.2,s,-2Pi,2Pi];,{s,-2Pi,2Pi,0.2}];   (* Animate!*)

3

f[x_] := Evaluate[Integrate[Cos[u^2/2], {u, 0, x}]];  f[x]

?FresnelC

Series[Sin[x], {x, 0, 11}]

Normal[Series[Sin[x], {x, 0, 11}]]

Plot[
      Evaluate[{Normal[Series[Sin[x], {x, 0, 11}]], 
          Normal[Series[Sin[x], {x, 0, 5}]], Sin[x]}], {x, -10, 
        10}]

Plot[f[x], {x, 0, 25}, PlotPoints -> 100];

Plot[f[x], {x, 0, 10}, PlotPoints -> 50];

g[x_, n_] := Normal[Series[f[x], {x, 0, n}]]

Plot[Evaluate[{f[x], g[x, 5]}], {x, 0, 10}, 
        PlotPoints -> 50];

Plot[Evaluate[{f[x], g[x, 30]}], {x, 0, 10}, 
        PlotPoints -> 50];

Plot[Evaluate[{f[x], g[x, 50]}], {x, 0, 10}, PlotPoints -> 50, 
        PlotRange -> {0, 2}];

g[x, 50]

Series[h[x], {x, 0, 50}]

4

Remove["Global`*"]

f[x_] := Sin[x];
Plot[f[x], {x, -Pi, 3  Pi}];

Series[f[x], {x, 0, 30}]

f1[x_] := Series[f[x], {x, 0, 30}] // Normal

Plot[Evaluate[{f1[x], f[x]}], {x, -5 Pi, 5  Pi}];

f2[x_] := 
        Series[f[x], {x, 0, 5}] // Normal;
Plot[Evaluate[{f2[x], f[x]}], {x, -5 Pi, 5  Pi}];

Join[{f[x]}, Table[f2[x, k], {k, -10, 10, 0.4}]]

f2[x_, k_] := 
        Series[f[x], {x, k, 1}] // Normal;
Plot[
        Evaluate[
          Join[{f[x]}, 
            Table[f2[x, k], {k, -10, 10, 0.4}]]], {x, -5 Pi, 
          5  Pi}];

Animate!

Table[
        Plot[Evaluate[{f[x], f2[x, k]}], {x, -5 Pi, 5  Pi}, 
          PlotRange -> {-2, 2}, 
          Epilog -> {PointSize[0.02], Point[{k, Sin[k]}]}], {k, -10, 
          10, 0.4}];